(IDF) rainfall curves in Senegal

30 Pages·2017·8.79 MB·English

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Intensity-Duration-Frequency (IDF) rainfall curves in Senegal
YoussouphSane1,GeremyPanthou2,AnsoumanaBodian3,TheoVischel2,ThierryLebel2,
HonoreDacosta4,GuillaumeQuantin2,CatherineWilcox2,OusmaneNdiaye1,AidaDiongue-Niang1,
andMarianeDiopKane1
1AgenceNationaledel’AviationCivileetdelaMétéorologie(ANACIM),Dakar,Sénégal
2Univ.GrenobleAlpes,IRD,CNRS,GrenobleINP,IGE,38000Grenoble,France
3LaboratoireLeidi,UniversitéGastonBerger,SaintLouis,Sénégal
4DépartementdeGéographie,UniversitéCheikhAntaDiop,Dakar,Sénégal
Correspondenceto:G.Panthou([email protected])
Abstract.Urbanizationresultingfromasharplyincreasingdemographicpressureandthedevelopmentofinfrastructureshave
made the populations of many tropical areas more vulnerable to extreme rainfall hazards. Characterizing extreme rainfall
distributioninacoherentwayinspaceandtimeisthusbecominganoverarchingneedthatrequiresusingappropriatemodelsof
IDFcurves.Using14seriesof5-minrainfallrecords(aggregatedatabasistime-stepof1h)collectedinSenegal,acomparison
5 of two GEV&scaling models is carried out, leading to adopt the most parsimonious one, built around four parameters. A
bootstrap approach is proposed to compute the uncertainty associated with the estimation of these 4 parameters and of the
related rainfall return levels for durations ranging from 1h to 24h. This study confirms previous works showing that simple
scalingholdsforcharacterizingthetime-spacestructureofextremerainfallintropicalregionssuchassub-SaharanAfrica.It
further provides confidence intervals for the parameter estimates, and shows that the uncertainty linked to the estimation of
10 the GEV parameters, is 3 to 4 times larger than the uncertainty linked to the inference of the scaling parameter. From this
model,mapsofIDFparametersoverSenegalareproduced,providingaspatialvisionoftheirorganizationoverthecountry,
withaNorthtoSouthgradientforthelocationandscaleparametersoftheGEV.Aninfluenceofthedistancefromtheocean
wasfoundforthescalingparameter.ItisacknowledgedinconclusionthatclimatechangerenderstheinferenceofIDFcurves
sensitivetoincreasingnonstationarityeffects,requiringtowarnend-usersthattheyshouldbeusedwithcareanddiscernment.
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1 Introduction
ThefastgrowingpressureofmankindonplanetEarthmakespopulationsincreasinglyexposedtohydrometeorologicalhazards
such as torrential rains and floods (IPCC, 2012; Mechler and Bouwer, 2015). Hydrologists are thus more compelled than
ever to deal with the problem of assessing the probability of extreme rainfall events at different time scales and for various
5 return periods, depending on the area of the target catchment and the issue at stake most notably human life protection and
infrastructuredimensioning.Aclassicalwayofsynthesizingtheresultsofsuchstudiesistheproductionofso-calledrainfall
Intensity-Duration-Frequency (IDF) curves, providing estimates of rainfall return levels over a range of durations. In doing
so, scientists are facing two sets of difficulties, one related to data availability and the other to the necessity of a proper
methodologicalframework.
10 Onthedataside,thefrequencyanalysisofextremesrequireslongandcontinuousrecordsofrainfallatthesamelocation,
somethingnotunusualatadailytime-stepeventhoughinsomeregionsthisisnotsocommon.Moreoveracomplicatingfactor
is that, in many cases, it is necessary to consider sub-daily time-steps, for which long term records of rainfall are much less
numerousormuchlessreliable/accuratethandailyseries.
Themethodologicalchallengearisesfromthecomplexcombinationoffactorscausingrainfalltobestronglyvariableatall
15 scales(fromthemicrophysicsdropletscalestosynopticscales),asaresultofthenon-linearinteractionofdifferentatmospheric
processes(e.g.SchertzerandLovejoy,1987).Thisimpliesthatitisnotatallobvioustofindapropertheoreticalframework
tocomputeIDFcurvesinawaythatensurescoherencybetweentimescales.EarlyworksonIDFproposedempiricalmethods
consisting of first adjusting a frequency distribution models fitted to rainfall series R(D) for each duration D of interest;
{ }
andthenIDFformula i (D) fittedindependentlytoeachseriesofquantilesderivedfromthefirststepandcorresponding
T
{ }
20 toagivenreturnperiod T (seee.g.Milleretal.,1973;NERC,1975).Thishastheadvantageofbeingeasilyimplementable
and is thus commonly used by hydrological engineers and operational climate/hydrological services. However, because of
uncertaintiesinthecomputationofthequantilesderivedforthedifferentdurations,thescalingformulationmaybephysically
inconsistentandmayleadtogrosserrorssuchasparasiticoscillationsorintersectionsbetweenIDFcurvescomputedfortwo
differentdurations(seeKoutsoyiannisetal.,1998,formoredetails).Asaremedytosuchinconsistencies,Koutsoyiannisetal.
25 (1998)werethefirsttoproposeageneralIDFformulationthatremainsconsistentwithboththefoundationsoftheprobabilistic
theories and the physical constraints of scaling across durations. Another notable advance was provided by Menabde et al.
(1999)whodemonstratedthatchangesinrainfalldistributionwithdurationformulatedbyKoutsoyiannisetal.(1998)canbe
expressedasasimplescalingrelationship,openingthepathforusingthefractalframeworkinordertodescribethetimescaling
betweenIDFcurvesestablishedoverarangeofdurationsinvariousregionsoftheworld(seee.g.Yuetal.,2004;Borgaetal.,
30 2005;GeroldandWatkins,2005;Nhatetal.;Baraetal.,2009;Blanchetetal.,2016;Rodríguez-Solàetal.,2016;Yilmazetal.,
2016).
Disposing of a consistent scaling framework does not suppress, however, the crucial sampling issues associated with the
estimation of the parameters of the IDF model. This involves significant uncertainties in the final determination of rainfall
returnlevels,aquestionrarelyaddressedintheliterature;onthataspect,seethepioneeringworkofMélèseetal.(2017)which
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presentandcomparedifferentmethodstocomputeIDFconfidenceintervals(onaGEV&scalingmodel)overtheMediterranean
region. It is especially important to investigate this issue in tropical regions such as Sub-Saharan Africa, where a number of
newinfrastructureprojectsareinthepipewhileatthesametimeasignificantincreaseoffloodrisksandrelatedmancasualties
hasbeenreportedoverthepast15yearsorso(Di-Baldassarreetal.,2010;Tschakertetal.,2010;IPCC,2014).Severalrecent
5 studies have dealt with the question of IDF calculation for different West African countries. Some focused on analysing the
behaviour of extreme rainfall distribution at a given location (such as Soro et al., 2008, 2010, for Ivory Coast) while others
lookedatthescalingbehaviouroverdurationssuchasMohymontandDemarée(2006)forCongoandOyegokeandOyebande
(2008)forNigeria.Van-De-VyverandDemarée(2010)alsoanalysedthescalingpropertiesofrainfalloverarangeofdurations
foracoupleofstationsinCongo,findingthevalueofthemainscalingparametertobelargerthantheoneobtainedforUcclein
10 Belgium,meaningthatthesmalldurationsareheavilydrivingthebehaviourofextremerainfallatlargerdurationsforatropical
climate.More recently,Panthouetal. (2014b)andAgbazoet al.(2016)showed thattheGEV&simple-scalingframework is
wellsuitedtoestimaterainfallreturnlevelsatvariousdurationsinacoherentwayforanarrayofstationscoveringamesoscale
areaoftypicallyadozenthousandskm2,respectivelyinSouth-WestNigerandinNorthernBenin.WhileAgbazoetal.(2016)
assume a Gumbel distribution of the annual maxima, Panthou et al. (2014b) used the approach in its broader formulation,
15 showingthattheannualmaximadistributionwasheavytailed(positivevalueoftheshapeparameteroftheGEV).Itisworth
notingthatinbothcasesusewasmadeofthehighqualityandfinetimescaleresolutiondatacollectedbytheAMMA-CATCH
researchobservatory(Lebeletal.,2009).Thisdatasetcoveringhomogeneouslyawiderangeoftime-steps(from5minutes
upward) over more than 20 years is unique in Tropical Africa. This means than in every other area, the parameters of the
scaling relationship will have to be inferred from a very limited number of sub-daily rainfall series, not all of them being of
20 same records length, thus raising the question of which parameters have the largest influences on the final uncertainties of
rainfallreturnlevels.Thisissueisextremelyimportantwhendealingwithlargeregions(suchasawholecountry)overwhich
the scaling parameters may vary spatially, making it unstraightforward to infer rainfall return levels for sub-daily durations
whenonlydailydataareavailable.FocusingonSenegal,aregionofcontrastedcoastaltoinlandsemi-aridclimate,ourpaper
ambitions are both to address the uncertainty issue not dealt with in above mentioned papers and to provide IDF curves for
25 aregionlocatedatthewesternedgeoftheSahel,lookingatthespatialvariabilitygeneratedbythetransitionfromthecoast
to inland. In addition to its methodological bearing, the paper aims at making these IDF curves widely accessible to a large
range of end-users in the whole country by mapping the values of the scaling parameters and of the rainfall return levels.
Furthermore,selectinganIDFmodelaslesssensitiveaspossibletodatasamplingeffectsandcomputingtheassociatedIDF
confidenceintervalsmakeeasiertheupdateoftheIDFcurveswhennewdataareavailable.
30 2 DataandRegion
2.1 SenegalClimatologicalContext
SenegalislocatedatthewesternedgeoftheAfricancontinentbetweenlatitudes12 Nand17 N(Figure1a).Theclimateof
◦ ◦
SenegalisgovernedbytheWestAfricanmonsoon(Laforeetal.,2011;Janicotetal.,2011;Nicholson,2013),resultingina
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two-seasonannualcycle:adryseasonmarkedbythepredominanceofmaritimeandcontinentaltradewindsinwinteranda
rainy season, marked by the progressive invasion of the West African monsoon (Figure 2a to Figure 2c) during the summer.
Thelengthoftherainyseasonvariesalongthelatitudeandrangesroughlyfrom5months(earlyofJunetotheendofOctober)
in the South to 3 months (mid-July to mid-October) in the Northern part of Senegal. Rainfall amounts peak in August and
5 September,coincidingwiththeperiodwhentheITCZreachesitsnorthernmostpositionoverSenegal.
ThereisastrongNorth-Southgradientofthemeanannualrainfall(Figure2d)rangingfrom300mmintheNorthtomore
than 1000 mm in the South (Diop et al., 2016). This gradient is mainly explained by the number of rainy days (in average
between20and80fromNorthtoSouth)andtoalesserextentbythemeanintensityofrainydays(inaveragebetween10and
15mm day 1),seeFigure2eand2f.
−
10 TherainsaremainlycausedbyMesoscaleConvectiveSystemssweepingthecountryfromeasttowest(Laurentetal.,1998;
Mathonetal.,2002;Diongueetal.,2002).SometimesitalsohappensthatcycloniccirculationsofftheSenegalesecoastdirect
moistureladenairflowoverthewesternpartofthecountry,dumpingheavyrainsthatoftencausefloodsincoastalcities.Due
toitsconvectivenature,rainfallisstronglyvariableinspaceandtime,especiallyattheeventscaleforwhichlargedifferences
inrainfallamountsarefrequentlyobservedintwoveryclosepoints(Saneetal.,2012).Theraindurationsarealsogenerally
15 short,exceptintherarecaseofstationaryconvectivesystems(blockedsituation).
Senegalregularlyundergoesdamaging heavydownpours.Arecentexampleisthe rainfalleventthatoccurredinDakar in
themorningofAugust26,2012,causingthelargestfloodoverthelasttwentyyearsinthecity.Anamountof160mmwas
recordedattheDakar-Yoffstationwhichislargebutnotahistoricalrecordatthisstation.Infact,thiseventwasexceptional
becauseofitsintensitiesatshortdurations(54mmwererecordedin15minutesand144mmin50minutes)exceedingby
20 farthepreviousrecordsinDakar-Yoff.Suchrainfallintensitiesandtheirassociateddisastersjustifytheimportanceofbetter
documentingextremerainfalldistributionsatshorttimescales.
2.2 RainfallData
Thearchivesofclimate/hydrologicalservicesofWestAfricancountriessometimescontainlargeamountofsub-dailyrainfall
records.Howevertheserecordsaremostofthetimestoredinpaperstripchartformatsrequiringatedioustaskofdigitization
25 forusingtheminnumericalapplications.
Thepresentstudyhasbeenmadepossiblethankstoanimportantworkofanalyzinganddigitizingrain-gaugechartscarried
out for the main synoptic stations of Senegal. This process was undertaken by the laboratory of hydro-morphology of the
GeographyDepartmentoftheUniversityCheikhAntaDiopofDakar(UCAD)incollaborationwiththeNationalAgencyof
CivilAviationandMeteorology(ANACIM)whoprovidedtherainfallpapercharts.
30 Senegalese synoptic stations are equipped with tipping bucket rain-gauges; the receiving ring is 400 cm2 and a bucket
corresponds to 0.5 mm of rain. The roll rotation is daily. The chart analysis has been performed with the software “Pluvio”
developedby(Vauchel,1992)allowingthecomputationof5-minutetime-stepdigitizedrainfallseriesfromthepaperdiagrams.
It is a long and laborious task, which has the advantage of allowing a careful “chart by chart” checking of the quality of
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the records before digitization. For more information on the digitization process, the reader may refer to the publications of
Laaroubi(2007)andBodianetal.(2016).
Atotalnumberof23tippingbucketrain-gaugeswasanalysed,withdatagoingbackto1955fortheoldestandto2005for
themostrecent.Astheassessmentofextremerainfalldistributionsisknownforbeingmuchsensitivetosamplingeffectand
5 erroneousdata(Blanchetetal.,2009;Panthouetal.,2012,2014b),aparticularattentionwaspaidtocheckandselectthemost
appropriateseries.
Thedataselectionhadtoconciliatetwoconstraints:(i)keepingthedatasetaslargeaspossibleand(ii)eliminatingseries
thatcontaintoomuchmissingdata.
Theprocedureforclassifyingoneyear-stationasvalidornotisthefollowing:(i)first,theannualnumberof5-mindataand
10 the annual amount of rain are computed, (ii) the mean inter-annual values of these two statistics are computed on the whole
series,(iii)ayearisclassifiedasvalidifeitherthenumberof5-minraindataortheamountofrainfalliscomprisedbetween1
/2.5and2.5timestheirmeaninter-annualvalues,(iv)otheryearsareclassifiedasmissingandremovedfromthewholeseries.
Since missing years influence the mean inter-annual values, step (ii), (iii) and (iv) are repeated until all remaining years are
classifiedasvalid(notethat,infact,nostation-yearhadtobeexcludedaftertheinitialstep).Allvalidyearsforallseriesare
15 plottedonFigure3.InordertokeeptheIDFfittingrobust,onlyserieswithatleast10yearsofvaliddatahavebeenused.This
ledustoretain14stationswithrecordlengthvaryingfrom10(Fatickstation)to44years(Ziguinchorstation)withamedian
of28years.Thisdatasethastheadvantageoffairlycoveringthewholecountrybutasthelengthoftheseriesvaries,thequality
oftheIDFestimatesmightdifferfromonestationtoanother.Thiseffectwillbemorepreciselyanalysedinsection5.2.1.
3 Theoreticalbackground
20 3.1 GeneralIDFformulations
3.1.1 EmpiricalIDFformulations
Intensity-Duration-Frequency(IDF)curvesprovideestimatesofrainfallintensityforarangeofdurations D andforseveral
{ }
frequenciesofoccurrence(usuallyexpressedasareturnperiodT).Eachcurvecorrespondstotheevolutionofareturnlevel
(i )asafunctionofrainfalldurationD.Historically,severalempiricalformulationsofIDFcurveshavebeenproposed.All
T
25 canbedescribedbythefollowinggeneralequation(Koutsoyiannisetal.,1998):
i (D)=w(T) [D+θ(T)]η(T) (1)
T
×
wherew,θandηareparameterstobecalibratedfromrainfallobservations.
3.1.2 Koutsoyiannisscalingrelationship
Koutsoyiannisetal.(1998)havedemonstratedthattheempiricalformulations(Equation1)canbeexpressedas:
30 iT(D)=a(T) bKoutso(D) (2)
×
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whereb (D)isthescalingfunction:
Koutso
b (D)=(D+θ)η (3)
Koutso
The advantage of Equation 2 as compared to Equation 1, is to separate the dependency on T (return period) from the
dependency on D (duration): a(T) only depends on T, and b (D) only depends on D. A consequence is that for the
Koutso
5 particularcaseofD0=1 θ
−
i (D )=a(T) (4)
T 0
Then,itbecomesaclassicalfrequencyanalysisoftherandomvariableI(D )toestimatethereturnlevelsi (D )-i.e.study
0 T 0
P[I(D0) i(D0)].Then,Equation2canbereformulatedasanequalityofdistributionofrandomvariablesI:
≤
d
I(D)=I(D ) b (D) (5)
0 Koutso
×
10 3.1.3 Simplescalingrelationship
Intheparticularcaseofθ=0,Equation5becomes:
d
I(D)=I(D ) b (D) (6)
0 SiSca
×
b (D)=Dη (7)
SiSca
15 wherebSiSca(D)isasimplescalingformulationofb.
3.2 IDFscalingformulationsintheframeoftheExtremeValueTheory
Inthescalingapproachdescribedabove,theestimationofrainfallreturnlevelsrequiresastatisticalmodelofrainfallintensity
distributionsinceequations5and6taketheformofanequalityofdistributions.Theextremevaluetheory(Coles,2001)isthe
mostcommonlyusedframeworkforderivingthesemodels.
20 3.2.1 BlockmaximaframeworkinExtremeValueTheory
The Extreme Value Theory (EVT) proposes two methods to extract samples of extreme values from a time series (Coles,
2001):theBlockMaximaAnalysis(BMA)whichconsistsofdefiningblocksofequallengths(oftenoneyearinhydrology)
and extracting the maximum value within each block; the Peak Over Threshold (POT) which consists of extracting all the
valuesexceedingagiventhreshold.
25 ComparedtoBMA,thePOThastheadvantageofallowingtheselectionofmorethanonevalueperyear,thusincreasing
thesamplesizeusedforinferringthemodel,butthechoiceofanappropriatethresholdisoftendifficult(Frigessietal.,2002).
HeretheBMAapproachwaspreferredasitismorestraightforwardlyimplementable.
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InBMA,whentheblockislargeenough(whichisensuredforannualmaxima),theExtremeValueTheorystatesthatthe
Generalized Extreme Value (GEV) distribution is the appropriate model for block maxima samples(Coles, 2001). The GEV
distribution is fully described by three parameters, the location (µ), the scale (σ), and the shape (ξ), which are respectively
relatedtotheposition,thespreadandtheasymptoticbehaviourofthetailofthedistribution:
1
i µ −ξ i µ
5 FGEV(i;µ,σ,ξ)=exp 1+ξ − for1+ξ − >0 (8)
(−(cid:20) (cid:18) σ (cid:19)(cid:21) ) (cid:18) σ (cid:19)
Apositive(negative)shapecorrespondstoaheavy-tailed(bounded)distribution.Whenξ isequalto0,theGEVreducesto
theGumbeldistribution(light-taileddistribution):
i µ
F (i;µ,σ)=exp exp − (9)
GUM
− − σ
(cid:26) (cid:20) (cid:18) (cid:19)(cid:21)(cid:27)
3.2.2 GEVparameterformulationinascalingframework
10 Menabde et al. (1999) have derived the equations merging the scaling formulations presented above (both bSiSca and bKoutso)
withtheextremevaluedistribution(seealsoPanthouetal.,2014b;Blanchetetal.,2016).Inthisapproach,theI(D)samples
aremodeledbyaGEVmodelforwhichthelocationandscaleparametersareparameterizedasafunctionofDasfollows:
I(D) GEV µ(D);σ(D);ξ (10a)
∼ { }
µ(D)= µ b(D) (10b)
0
×
15 σ(D)= σ0 b(D) (10c)
×
ThereturnlevelsareeasilyobtainedatalldurationsDas:
1
iT(D)=FG−E1V D,1−T (11)
(cid:18) (cid:19)
Thisformulationisequivalenttothefollowing:
1
20 iT(D)=FG−E1V D0,1−T ×b(D) (12)
(cid:18) (cid:19)
WithD =1 θ.
0
−
Notethatequations10to12arevalidforbothb andb .
SiSca Koutso
4 Methodology:inference,evaluationanduncertaintyofIDFmodels
In this study, two IDF models are compared: the IDF obtained from the Koutsoyiannis scaling b and the IDF
Koutso Koutso SiSca
25 obtained from the simple scaling bSiSca. Both models describe the distribution of extreme rainfall intensities across dura-
tion but they differ in their formulation and in their number of parameters: IDF has four parameters µ ,σ ,ξ,η while
SiSca 0 0
{ }
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IDF has five parameters µ ,σ ,ξ,η,θ .The BMA samples from which the two scaling models are inferred and eval-
Koutso 0 0
{ }
uated are built by using 1-year block lengths, in order to ensure the independency between the elements of the sample. At
eachstation,theextremerainfallsamplethusconsistsofannualmaximumintensitiesi(D)withD rangingfrom1hto24h:
1,2,3,4,6,8,10,12,15,18,24 h.Notethat,foreachdurationD,arollingmeanoflengthDisappliedtothe5minuterainfall
{ }
5 seriesbeforeextractingthemaxima.Thisensuresthattheextractedmaximaarenotunder-estimated(whichisthecasewhen
usingafixedwindow).
4.1 IDFmodelinference
Different fitting methods have been tested to adjust the IDF model parameters to rainfall data: one of them (the two-step
method)isapplicabletobothIDF andIDF models.
Koutso SiSca
10 NotethattwoothermethodsspecificallydedicatedtotheIDFSiSca modelhavealsobeentested:onebasedonthemoment
scaling function (as in e.g. Borga et al., 2005; Nhat et al.; Panthou et al., 2014b), and one based on the global maximum
likelihood estimation (as in Blanchet et al., 2016). As they did not perform better than the two-step method, they are not
presentedhere.
Thefittingofthescalingb(D)isbasedontheequalityofdistributiongiveninEquation5forIDF andEquation6for
Koutso
15 IDFSiSca.Iftheseequationshold,thescaledrandomvariablesI(D)/b(D)haveforalldurationsDthesamedistributionasthe
random variable I(D ). This means that the observed scaled samples i(D)/b(D) have similar statistical properties for each
0
durationD.Basedonthisproperty,theparametersofb(D)arecalibratedinordertominimizeastatisticaldistancebetweenthe
differentscaledsamplesi(D)/b(D).AssuggestedbyKoutsoyiannisetal.(1998),thedifferenceinmedianscomputedbythe
Kruskal-Wallisstatisticappliedonmulti-samples(KruskalandWallis,1952)waschosentocharacterizethedistancebetween
20 thescaledsamplesi(D)/b(D).
Oncethescalerelationshipisidentified,scaledsamplesi(D)/b(D)arecomputedandpooledinasinglesamplesincethey
are expected to follow the same GEV distribution (see Equation 12). The GEV parameters are estimated on this aggregated
scaledsamplebyusingtheL-Momentsmethod.Thismethodwasretainedasitismoresuitableforsmallsamples(Hosking
andWallis,1997)thantheMaximumLikelihoodEstimationalgorithmwhichsometimesfailsinoptimizingthelikelihoodfor
25 toosmallsamples.
4.2 Modelsevaluationandselection
WiththeaimofselectingthebestIDFmodelfromthetwocomparedIDFformulations(IDF andIDF ),aprocessof
Koutso SiSca
model evaluation and comparison is proposed here by looking at both their flexibility (the models are fitted on a calibration
sample)andtheirrobustness(themodelsarefittedinapredictivemodeonsamplesnotusedforcalibration).
30 Theflexibilitycharacterizesthecapacityofamodeltofittheobserveddatawhichareusedtocalibrateitsparameters.To
thatpurpose,theIDFmodelsarefittedateachstation;thendifferentscoresarecomputedtoassessthefittingperformances.
The robustness, on the other hand, aims at evaluating whether the flexibility is not overstretched due to the model having
too many parameters with respect to the number of observations. As the two models tested here have a different number of
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parameters(4forIDF ,5forIDF ),thereisaparticularinterestatcomparingbetweenmodelshowthegoodnessof
SiSca Koutso
fitisdegradedwhenshiftingfromthecalibrationmodetothepredictivemode.ThepredictivecapacityoftheIDFmodelsis
assessed by using a classical calibration/validation process. At each station, a subset of data is used to fit the IDF model; a
secondindependentsubsetisusedtovalidateit.Thesamescoresasinthecalibrationmodearecomputedforthevalidation
5 subset. Rather than using two consecutive sub-periods, one for the calibration sample and one for the predictive sample, a
yeartoyearseparationwasusedtobuildthetwosubsets.Thislimitstheriskofobtainingsamplesmadeofyearsbelonging
predominantlytoadryperiodortoawetperiod.
TheflexibilityandthepredictivecapacityoftheIDFmodelsarequantifiedbasedontwotypesofscores:globalandquantile-
quantile.
10 The two global scores are the statistics returned by two goodness of fit (GOF) tests: Kolmogorov-Smirnov (KS) and
Anderson-Darling (AD). Each test computes a statistic based on the differences between a theoretical Cumulative Distri-
butionFunction(CDF)andtheempiricalCDF.Thenullhypothesisisthatthesampleisdrawnfromthefittedmodel.Thetest
returnsalsothecorrespondingp-value(erroroffirstkind).Thep-valueisusedasanacceptation/rejectioncriterionbyfixinga
threshold(here1%,5%,and10%).Thesetestsandp-valueswerecomputedforeachrainfalldurationateachstation.
15 GOFtestsallowevaluatingthewholedistributionbutdonotguaranteethatallquantilesarecorrectlyestimated.Thus,asa
complement,quantilebasedscoresarealsocomputed.Theycharacterizetherelationshipbetweentheoretical(obtainedfrom
the fitted CDF) and empirical quantiles (obtained from the empirical CDF). The root mean square error (RMSE), the mean
error(ME),andthemeanabsoluteerror(MAE)quantile-basedscoresarecomputed.Thefullpresentationofthesesscorescan
befoundinPanthouetal.(2012).Aweightedversionofthesescoresisalsousedinordertoassigngreaterweighttounusual
20 quantiles,asproposedbyBegueriaandVicente-Serrano(2006)andalsopresentedinPanthouetal.(2012).
4.3 Uncertaintyassessment
Fromamethodologicalpointofview,thecentralcontributionofthispaperisitsattemptatquantifyingtheuncertaintyasso-
ciated with IDF calculation in a scaling framework. This involves two distinct aspects. One is the uncertainty linked to the
estimationofthescalingparameters.TheotheristheuncertaintylinkedtotheinferenceoftheGEVparameters.Thissecond
25 componentisespeciallyimportanttoconsiderwhenapplyingascalingmodeltoalocationwheredailyrainfallseriesonlyare
available,whichistheultimatepurposeofregionalIDFmodels.Indeedinsomeregionalstudies,thescalingparameterswill
havetobeinferredfromaveryfewstationswhererainfallisrecordedatsubdailytime-steps;iftheydisplayvariationsinspace,
theythenwillhavetobespatiallyinterpolatedsoastoprovidescalingparameteratanylocationofinterest,notablyattheloca-
tionofdailyrainfallstations.Atthesestations,thescaledGEVdistributionisthusestimatedfromthedailyobservationsonly,
30 makingtheinferencefarlessrobustthanwhenusingaricherscaledsampleobtainedfromobservationsrangingfromonehour
-orless-tooneday.
Therefore, in the following, the uncertainty assessment at a given location will be addressed separately for the two situa-
tions: i) first when observations at this location are available over a whole range of time-steps; ii) secondly when only daily
observationsareavailable.
9
Nat.HazardsEarthSyst.Sci.Discuss.,https://doi.org/10.5194/nhess-2017-352
ManuscriptunderreviewforjournalNat.HazardsEarthSyst.Sci.
Discussionstarted:24October2017
c Author(s)2017.CCBY4.0License.
(cid:13)
4.3.1 Uncertaintylinkedtotheinferenceofthescalingmodelatlocationswithmultitime-scaleobservations
ConfidenceintervalsforIDFparametersandreturnlevelsareestimatedusinganon-parametricbootstrap(EfronandTibshirani,
1994). For each station, it consists of fitting IDF curves to bootstrap samples (i(D) ) obtained from the original i(D)
boot
samples.Theentireprocessconsistsofthreesteps:
5 1. The vector of years is resampled with replacement (Monte Carlo resampling) until its length equals the length of the
originalvector.
2. Onceayearyisdrawninthebootstrapsampleofyears,theannualmaximumforthatyearisretainedforeachdurationD
inordertobuildthebootstrapsampleofrainfallintensitiesi(D) .Thisguaranteesthecoherencebetweenthesamples
boot
atdifferentdurations.
10 3. TheIDFmodelisfittedonthebootstrapsamplei(D)boot.
4. Theobtainedparameters- µ ,σ ,ξ,η,θ forIDF and µ ,σ ,ξ,η forIDF -andtheassociatedreturn
0 0 boot Koutso 0 0 boot SiSca
{ } { }
leveli (D) arestored.
T boot
Thesefourstepsarerepeated1000timesleadingtogenerate1000i(D) samplesandobtainedvectorsoflength1000for
boot
thedifferentIDFparametersandforthedifferentIDFreturnlevelsstoredinstep4.Confidenceintervalsarecomputedonthese
15 vectors.Itisimportanttounderlinethattheseconfidenceintervalsareameasureoftheglobaluncertaintyassociatedwiththe
inferenceIDFmodel(uncertaintyduetotheinferenceofthescalingrelationshipanduncertaintygeneratedbytheinferenceof
theparametersofthescaledGEV).
4.3.2 GEVversusdonwscalingsourcesofuncertaintyatlocationswhereonlydailyobservationsareavailable
When only daily observations are available, the GEV parameters are inferred on the corresponding annual block maxima
20 sample of daily data, which contains far less information that the scaled samples used for fitting a scaled GEV when multi
timescaleobservationsareavailable.TheGEVparametersforthesub-dailytime-stepsarethendeducedfromthedailyGEV
parametersusingscalingparametersthatmustbeinferredfromnearbymultitimescaleobservations.Insomecasesthismight
generate a significant departure from the GEV model that would have been fitted directly on the observations at the proper
time-stepiftheywereavailable.Thiseffectisstudiedherebyassumingthatonlythedailydatawereavailableforfittingthe
25 GEVatour14stationsandimplementingthebootstrapapproachinawaythatallowsseparatingtheuncertaintylinkedtothe
GEVparameterinferenceandtheuncertaintylinkedtotheinferenceofthescalingparameters.Thisinvolvestwoindependent
bootstrapresamplingprocesses.
Thefirstconsistsinresamplingi(24h)basedon1000bootstrapdrawingsandfitting1000GEV(24h)tothesebootstrapsam-
ples.These1000GEV(24h)arethendownscaledtoatargetdurationD usingEquation10,yielding1000differentGEV(D)
30 (practicallyonlytheresultsobtainedforthe1-hourdurationarepresentedhere).ThescalingparametersusedtoinformEqua-
tion10arethosecomputedfromthecompletemultitime-stepdatasetasexplainedinsection4.1.Thisprocessyieldsasample
10