Testing the Beta-Lognormal Model in Amazonian Rainfall Fields Using the Generalized Space q ...

25 Pages·2017·2.04 MB·English

entropy
Article
Testing the Beta-Lognormal Model in Amazonian
Rainfall Fields Using the Generalized
Space q-Entropy
HernánD.Salas* ID,GermánPoveda ID andOscarJ.Mesa ID
FacultaddeMinas,DepartamentodeGeocienciasyMedioAmbiente,UniversidadNacionaldeColombia,
SedeMedellín,Carrera80#65-223,Medellín050041,Colombia;[email protected](G.P.);
[email protected](O.J.M.)
* Correspondence:[email protected]
Received:9October2017;Accepted:8December2017;Published:13December2017
Abstract:WestudyspatialscalingandcomplexitypropertiesofAmazonianradarrainfallfieldsusing
theBeta-LognormalModel(BL-Model)withtheaimtocharacterizeandmodeltheprocessatabroad
range of spatial scales. The Generalized Space q-Entropy Function (GSEF), an entropic measure
definedasacontinuoussetofpowerlawscoveringabroadrangeofspatialscales,S (λ) ∼ λΩ(q),
q
is used as a tool to check the ability of the BL-Model to represent observed 2-D radar rainfall
fields. Inaddition,weevaluatetheeffectoftheamountofzeros,thevariabilityofrainfallintensity,
thenumberofbinsusedtoestimatetheprobabilitymassfunction,andtherecordlengthontheGSFE
estimation. Ourresultsshowthat: (i)theBL-Modeladequatelyrepresentsthescalingproperties
of the q-entropy, S , for Amazonian rainfall fields across a range of spatial scales λ from 2 km to
q
64km;(ii)theq-entropyinrainfallfieldscanbecharacterizedbyanon-additivityvalue,q ,atwhich
sat
rainfall reaches a maximum scaling exponent, Ω ; (iii) the maximum scaling exponent Ω is
sat sat
directlyrelatedtotheamountofzerosinrainfallfieldsandisnotsensitiveto eitherthenumber
ofbins toestimatetheprobabilitymassfunctionorthevariabilityofrainfallintensity;and(iv)for
small-samples,theGSEFofrainfallfieldsmayincurinconsiderablebias. Finally,forsynthetic2-D
rainfallfieldsfromtheBL-Model,welookforaconnectionbetweenintermittencyusingametric
basedongeneralizedHurstexponents, M(q ,q ),andthenon-extensiveorder(q-order)ofasystem,
1 2
Θ ,whichrelatestotheGSEF.Ourresultsdonotexhibitevidenceofsuchrelationship.
q
Keywords: hydrology;tropicalrainfall;statisticalscaling;Tsallisentropy;multiplicativecascades;
Beta-Lognormalmodel
1. Introduction
1.1. StatisticalScalingandMultiplicativeRandomCascades
Statisticalscalinghasprovidedarichframeworktounderstandandmodelthespatiotemporal
dynamicsandthecomplexityandintermitencyofrainfallfields,including(multi-)fractal,multiscaling,
andrandomcascademodels[1–19].
Thestrongvariabilityandintermittenceofconvectivetropicalrainfallconstituteanadequate
settingtostudythescalingcharacteristicsofrainfallinawiderangeofspatio-temporalscales[19–31].
Inparticular,Ref. [19]foundthat2-DrainfallfieldsoverAmazoniaexhibitmultiscalingpropertiesin
space,whichmeansthattherelationship M (λ) ∼ λ−τ(r) exhibitsanon-linearbehavior,whereλis
r
thespatialscale,rtheorderofthestatisticalmomentandτ(r)isther-thmomentscalingexponent.
Additionally, they show that both the diurnal cycle and the predominant atmospheric regime of
Amazonianrainfall(EasterlyorWesterly)exertastrongcontrolonthescalingpropertiesofAmazonian
Entropy 2017,19,685;doi:10.3390/e19120685 www.mdpi.com/journal/entropy
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storms,thussheddinglighttowardsunderstandingthelinkagesbetweenscalingstatisticsandphysical
featuresofAmazonianrainfallfields.
1.2. MultiplicativeRandomCascadesandtheBeta-LogNormalModel
TheBeta-LognormalModel(hereafterBL-Model)isadiscrete2-Drandomcascadenon-Markovian
model[13]basedontheobservedscalingpropertiesofrainfall,withonlytwoparameters: σdenoting
thevariabilityofrainfallintensity,and βrepresentingtherainyareafraction. Thismodelprovides
aframeworktocarryoutnumericalexperimentscontrollingfeaturesofrainfall(e.g.,σandβ),butalso
to link diverse statistical and physical characteristics across spatial scales (e.g., Refs. [19,31] and
Section1.5, respectively). Inaddition, themodelprovidesatooltoinvestigatetherobustnessand
sensitivityofstatisticalmetricstopoorsampling,datasparsityandintermittencyofhigh-resolution
2-Drainfallfields.
Theconstructionofaspatiallydistributeddiscreterandomcascademodelusuallybeginswith
agivenmass(orvolume)ofrainfalloveratwo-dimensional(d =2)boundedregion[6]. Theregion
issuccessivelydividedintobequalparts(b = 2a)ateachstep,andduringeachiterationthemass
obtainedatthepreviousstepisdistributedintothebsubdivisionsthroughmultiplicationbyasetof
“cascadegenerator”W,asshownschematicallyinFigure1(forthecaseofd = 2andb = 4). Ifthe
initialarea(atlevel0)isassignedanaverageintensityR ,thisgivesaninitialvolumeR Ld,whereL
0 0 0 0
istheouterlengthscaleofthestudyarea. Thus,atthefirstlevelthevolumeissubdividedintob =4
subareasdenotedby∆i,i = 1,2,...,4. Atthesecondlevel,eachoftheprevioussubareasisfurther
1
subdividedintob =4subareas,whicharedenotedby∆i,i=1,2,...,16,foratotalofb2 =16subareas.
2
Thissubdivisioniscontinuedfurtherdownthespatialscale,leadingatthenthlevel,tobn subareas
denotedby∆i,i=1,2,...,bn.
n
AsshowninFigure1,afterthefirstsubdivision,thefoursubareas(∆i,i =1,2,...,b)areassigned
1
volumes R Ld b−1 Wi,fori = 1,2,...,b. Uponsubdivision,thevolumesµ ∆i insubareasatthenth
0 0 1 n 1
subdivision,∆i,i=1,2,...,bn,aregivenby,
n
n
µ ∆i = R Ldb−1∏Wi, (1)
n 1 0 0 j
j=1
where,foreachcascade’slevelj,irepresentsonesubareabelongingtothelevel. ThemultipliersW in
Equation(1)arenon-negativerandomcascadegenerators,withE[W] =1toensurethatthemassis
conservedonaverage,fromonediscretizationleveltothenextone. OverandGupta[13,32]proposed
the so-called BL-Model for the cascade generators W. The BL-Model considers W as a composite
generator,W = BY,whereBisageneratorfromthe“Betamodel”andYisdrawnfromaLognormal
distribution [33]. Essentially, the Beta model partitions the region into sets with and without rain,
while the Lognormal model then assigns a certain amount of rainfall to each rainy area fraction.
TheBetamodelexhibitsadiscreteprobabilitymassfunctionwithjusttwopossibleoutcomes(B =0
andB = bβ),givenas
P(B =0) =1−b−β P(B = bβ) = b−β, (2)
wherebisthebranchingnumberandβisaparameter. SinceYbelongstotheLognormaldistribution,
itcanbeexpressedasY = b−σ2l2n(b)+σX,whereXisastandardNormalr.v. andσ2(>0)isaparameter
equaltothevarianceoflog Y,withtheconditionthatE[Y] =1. Insuchcase,itiseasytoshowthat
b
theconditionE[W] =1isalsosatisfied. TheprobabilitydistributionfunctionofW = BYcanthusbe
expressedas
P(W =0) =1−b−β, (3)
P(W = bβY = b−σ2l2n(b)+σX) = b−β. (4)
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The parameters of the BL-Model (β and σ2) can be estimated [13,32] through the so-called
Mandelbrot–Kahane–Peyriere (MKP) function [34,35]. The MKP function characterizes the fractal
orscale-invariantbehaviorofthemultiplicativecascadeprocess. OverandGupta[32]theoretically
derived an expression for χ(r) for the BL-Model, in terms of the cascade parameters β, σ2, b and
exponentr,suchthat,
σ2ln(b)
χ (r) = (β−1)(r−1)+ (r2−r). (5)
b
2
Thus,providedthatarainfallfieldbelongstoadiscreterandomcascadewithgeneratorssatisfying
theBL-Model,theexpressiongiveninEquation(5)canbematchedwiththeempiricallydetermined
estimators,τ(r)/d,toestimateβandσ2. Thefirstandsecondderivativesofτ(r) = dχ(r)withrespect
tor,thelatterofwhichisgivenbyEquation(5),canbeusedtoobtain[20,32],
(cid:20) σ2ln(b) (cid:21)
τ(1)(r) = d β−1+ (2r−1) , (6)
2
(cid:104) (cid:105)
τ(2)(r) = d σ2ln(b) (7)
Both τ(1)(r) and τ(2)(r) can be computed by numerically estimating the derivatives of the
empirical slopes of the scaling relation between τ(r) and r, using the log–log plotting of M(λ ,r)
n
versusλ as,
n
M(λ ,r) = [λ ]τ(r), (8)
n n
whereM(λ ,r)arethesamplemomentsM(λ ,r)andλ isthecorrespondingscaleratio.Equations(6)
n n n
and (7) are combined together to express the cascade parameters β and σ2 in terms of τ(1)(r) and
τ(2)(r)asfollows:
τ(1)(r) σ2ln(b)
β =1+ − (2r−1), (9)
d 2
σ2 = τ(2)(r)/dln(b) (10)
Equations(9)and(10)areevaluatedforagivenvalueofr. Theusualpracticeistouser = 1,
although[6]usedr =2fortestingaspace-timemodelofdailyrainfallinAustralia.
λ=1
n=0
Wo
λ=1/2
n=1 WoW1
λ=1/4
n=2
WoW1W2
Figure1.SchematicplotoftherandomcascadegeometrytakenfromGuptaandWaymire[33].
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1.3. q-Entropy
Thereisawidefamilyofgeneralizedentropicfunctionswithvariousdegreesofsophistication
intheliterature[36–40]. Inparticular, Tsallis[40]proposedtheconceptofnonadditiveentropy S
q
(hereafterq-entropy),whichhasshowntobeusefulinthestudyofabroadrangeofphenomenaacross
diversedisciplines[41,42],relatedtothewellknownRényientropy[43]. The q-entropyisdefinedas,
S = 1−∑in=1pq(xi) (cid:18)∑n p(x ) =1; q ∈ (cid:60)(cid:19). (11)
q q−1 i
i=1
which, inthelimit q → 1, recoverstheusualBoltzmann–Gibbs–Shannonentropy, S [44], whichis
additive;inotherwords,forasystemcomposedofanytwo(probabilistically)independentsubsystems,
theentropySofthesumisthesumoftheirentropies[45],suchthat,if AandBareindependent,
Sq=1(A+B) = S(A+B) = S(A)+S(B). (12)
ItturnsoutthatTsallisentropy,S (q (cid:54)=1),violatesthisproperty,andisthereforenonadditive.Thus,
q
theadditivitydependsonthefunctionalformoftheentropyintermsofprobabilities[45]. Therefore,
if AandBareindependent,then
S (A+B) = S (A)+S (B)+(1−q)S (A)S (B). (13)
q q q q q
Moregenerally,if AandBarenotprobabilisticallyindependentthen,
S (A+B) = S (B)+S (A|B)+(1−q)S (B)S (A|B). (14)
q q q q q
Taking the words of Tsallis [45], the value of q is useful to characterize the universal classes of
nonadditivity. He argues that it is determined a priori by the microscopic dynamics of the system,
whichmeansthatthethermostatisticalentropyisnotuniversalbutdependsonthesystemor,more
precisely,onthenon-additiveuniversalityclasstowhichthesystembelongs. Itisworthmentioning
thedifferencebetweenadditivityandextensivity,whichisclearlyexplainedin[45],asfollows:
“Anentropyofasystemorofasubsystemissaidextensiveif,foralargenumberNofitselements
(probabilisticallyindependentornot),theentropyis(asymptotically)proportionaltoN. Otherwise,
itisnonextensive.Thismeansthatextensivitydependsonboththemathematicalformoftheentropic
functionalandthepossiblecorrelationsexistingbetweentheelementsofthesystem. Consequently,
fora(sub)systemwhoseelementsareeitherindependentorweaklycorrelated,theadditiveentropy
Sisextensive,whereasthenonadditiveentropyS (q (cid:54)=1)isnonextensive. Incontrast,however,
q
fora(sub)systemwhoseelementsaregenericallystronglycorrelated,theadditiveentropyScanbe
nonextensive,whereasthenonadditiveentropyS (q (cid:54)=1)canbeextensiveforaspecialvalueofq.”
q
Additionally,recentstudies [28,30,31]haveinvestigatedtherelationshipbetweentheq-entropy
andtheGeneralizedParetodistribution(whichisrelevantinhydrologicalanalysis). Inparticular,
themaximizationoftheq-entropyunderaprescribedmeanleadstoaParetoprobabilitydistribution
withpower-lawtail[28,46–48],whichbelongstothefamilyofLévystabledistributions[49],specifically
toTypeIIGeneralizedParetodistributions. For1< q <2,theoriginaldistributiontakestheformof
theZipf–Mandelbrottype[50–52],whichdecaysasapowerlawforlargevaluesofx,andallmoments
aredivergentwhen3/2< q <2[53].
As such, the q-entropy constitutes a useful tool for the characterization of rainfall, and at the
sametimemotivatesinterestingdiscussionsaboutthephysicalinterpretationofentropicnon-linear
metrics,theirconnectionwithstochasticprocesses(MultiplicativeCascades),andallowsrevisitingits
applicabilityingeosciences(seeSection5).
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1.4. GeneralizedSpace-Timeq-EntropyinRainfallData
Previousstudies[28,30,31]introducedtheGeneralizedSpace-Timeq-Entropyasanewmethod
tostudytheorganizationdegreeandthescalingpropertiesofrainfall,byconsideringthespace-time
structure of rainfall as a system conformed by correlated subsystems, which is evident in the
hierarchicalstructureofconvectiverainfall. Thetimegeneralizedq-entropyisdefinedasafunctionof
order,q,andaggregationinterval,T,as[28]:
S (T) = 1−∑in=1pq(xi,T) (cid:18)∑n p(x ) =1; q ∈ (cid:60)(cid:19), (15)
q q−1 i
i=1
whereqisthestatisticalorderandP(x,T)istheprobabilityofoccurrenceofx ataggregationinterval,
i i
T. Ever since the original work [28], different characteristics of the Space and Time Generalized
q-entropyofrainfallwerereportedlateron[30,31],suchas:
• S (T)decreasesmonotonicallywithqforallvaluesofT.
q
• Foragivenvalueofq,estimatesareinverselyrelatedtoTforq <0,butdirectlyrelatedforq ≥0.
• EstimatesofSq(T) |q=1recoverthestandardentropyfordifferentvaluesofT.
• EstimatesofS (T)increasewithTforvaluesofq ≥0,uptoacertainsaturationvalue(maximum
q
q-entropy).
• ThefunctionS(T)vs. T inlog–logspace,fordifferentvaluesofq,canbeconsideredan(time)
entropyanalogousofthe(space)structurefunctioninturbulence[54].
• Thescalingexponents,Ω(q),ortheslopeoftherelationS(T)vs. Tinlog–logspace,fordifferent
valuesofq,exhibitanon-lineargrowthwithq,suchthatΩ ≈0.5forq ≥1. Thisresultallowed
sat
extendingtheconclusionsfromthestandardShannonentropytothegeneralizedq-entropy.
• Thescalingexponentsofsaturation,Ω ,aredifferentintimeandspaceforhydrologicaldata,
sat
such as time series of rainfall, for which Ω (q > 1.0) = 0.5, for time series of streamflows,
sat
forwhichΩ (q > 1.0) = 0.0,andforthespatialanalysisofradarrainfallfieldsinAmazonia,
sat
forwhichΩ (q >1.0) =1.0.
sat
AnalogoustoEquation(15),theSpaceGeneralizedq-Entropywasintroducedby[31],tostudy2-D
rainfallfields,usingλasthespatialscale,andP(x,λ)theprobabilityofoccurrenceofx associated
i i
withλ,suchthat
S (λ) = 1−∑in=1pq(xi,λ) (cid:18)∑n p(x ) =1; q ∈ (cid:60)(cid:19). (16)
q q−1 i
i=1
Withtheaimoflinkingthepresentstudywiththepreviousones, someimportantresultsare
worthmentioning:
• Poveda[28]studiedthetimescalingpropertiesoftropicalrainfallintheAndesofColombiaupon
temporalaggregationandintroducedtheGeneralizedTimeq-EntropyFunction(GTEF),asatime
analogousforq-entropyofthestructurefunctioninturbulence[54]. Heshowedthatthescaling
exponents, Ω(q) of the relation S (T) vs. T in log–log space, for different values of q, exhibit
q
anon-lineargrowthwithquptoΩ =0.5forq ≥1,puttingforwardtheconjecturethatthetime
dependentq-entropy,S (T) ∼ TΩ(q) withΩ(q) (cid:39)0.5,forq ≥1.
q
• SalasandPoveda[30]revisitedresultsreportedin[28],andanalyzedthetimescalingproperties
ofShannon’sentropyforthesamedatasetintermsofthesensitivitytotherecordlength,andthe
effectofzerosinrainfalldata,andproposedtheGTEFtostudythescalingpropertiesofriver
flows. Theyhighlightedtwoimportantresults:(i)ThescalingcharacteristicsofShannon’sentropy
differbetweenrainfallandstreamflowsowingtothepresenceofzerosinrainfallseries;and(ii)the
GTEFexhibitsmulti-scalingforrainfallandstreamflows. Forrainfall,therelationS (T)vs. T
q
inlog–logspacefordifferentvaluesofq,exhibitsanon-lineargrowthwithq,uptoΩ =0.5for
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q ≥1,incontrasttothescalingpropertiesofriverflowswhichexhibitanon-lineargrowthwithq,
uptoΩ =0.0forq ≥1.
• Poveda and Salas [31] studied diverse topics such as statistical scaling, Shannon entropy and
Space-TimeGeneralizedq-EntropyofMesoscaleConvectiveSystems(MCS)asseenbytheTropical
Rainfall Measuring Mission (TRMM) over continental and oceanic regions of tropical South
America, and in Amazonian radar rainfall fields. The main result of their study is that both
the GTEF and GSEF exhibit linear growth in the range −1.0 < q < −0.5, and saturation of
the exponent Ω for q ≥ 1.0, but for the spatial analysis (GSEF) the exponent tappers off at
sat
(cid:104)Ω (cid:105) ∼1.0,whereasforthetemporalanalysis(GTEF)theexponentsaturatesat (cid:104)Ω (cid:105) = 0.5.
sat sat
Inaddition,resultsaresimilarfortimeseriesextractedfromradarrainfallfieldsinAmazonia
(radarS-POL)andin-siturainfallseriesinthetropicalAndes.
1.5. EasterlyandWesterlyRegimesofAmazonianRainfall
TheWETAMC/LBAcampaign(January–February1999)foundthatwet-seasonconvectionin
Amazoniaexhibitstwogeneralmodes,hereinafter,theWesterlyandEasterlyregimes[55],whichare
highlycorrelatedtochangesinthe850–700hPazonalwinddirection[56]. Accordingtodatafrom
theNCEP-NCARReanalysis,aswellasfromtheFazendaNossaSenhoraradiosonde,theEasterly
(negativevalues)andWesterly(positive)regimesareclearlydifferentiatedinbothdatasets[19,31].
Furthermore, radar observations from southwestern Amazonia during TRMM-LBA suggest that
therewasrelativelylittledifferenceinthedailymeanrainfalltotalsbetweenEasterlyandWesterly
regimes [57]. In addition, the TRMM-LBA and TRMM satellite observations suggested marked
differencesinrainfallratedistributions,withtheEasterlyregime(Altiplano/southernBrazil)associated
withabroaderrain-ratedistributionandgreaterinstantaneousrainfallrates[56]. Precipitationfeatures
duringbothregimescanbesummarizedconsideringthatduringtheEasterlyregimeatmospheric
conditionsarerelativelydry,withincreasedlightningactivityandmoreintenseanddeeperconvective
systems. Incontrast,theWesterlyregimeischaracterizedbyadiminishedlightningactivity,lessdeep
convectionandlessintenseprecipitationrates[56,58,59]. Theregimeassociatedwithstrongervertical
development,morelightningactivity,andlargerinstantaneousrainfallratesmustbeassociatedwith
amore“concentrated”dailylatentheatrelease. Inaddition,twomechanismshavebeenproposedto
explaintheobservedchangesintheoverallconvectivestructureandlightningfrequencybetweenthe
tworegimes,thesemechanismsaretiedtoeitherthermodynamics(changesinCAPEandCINthat
modifytheenergeticsofthecloudensemble),oraerosolloading(e.g.,changesincloudcondensation
nuclei(CCN)concentrationthatmodifymicrophysicalstructureofthecloudensemble)[56].
Theaforementionedresultshaveimportantpracticalimplicationsinthespatialandtemporal
scaling features of rainfallfields [28,30,31] although theconnections between entropic and scaling
statistics with physical characteristics remain elusive. Furthermore, the effect of the space-time
structureofrainfallinthescalingoftheq-Entropyaswellasitssensitivitytothenumberofbinsandto
thevariabilityofrainfallintensityandthesample-sizemustbeinvestigatedindepth. Therefore,inthe
presentstudy,weaimtoinvestigatehowthespatialscalingandcomplexityofrainfallisreflectedin
differententropicscalingmeasureswithintheframeworksofinformationtheoryandnon-extensive
statisticalmechanics. Therationaleandobjectivesofthisworkarepresentednext.
1.6. RationaleandObjectives
Theobjectivesofourstudyarebaseduponthefollowingconsiderations:
• Thepresenceofzerosinhighresolutionrainfallrecordsconstitutehighlyimportantinformation
tounderstand,diagnoseandforecastthedynamicsofrainfall[28,60,61]. SalasandPoveda[30]
arguedthatzeros(inter-stormperiods)intimeseriesoftropicalconvectiverainfallareassociated
withthetimescalerequiredbynaturetobuildupthedynamicandthermodynamicconditionsof
thenextstorm,asanatmosphericanalogousofthetimeofenergybuild-upbetweenearthquakes,
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avalanches and many other relaxational processes in nature [62]. Therefore, the role of zeros
andtheireffectonscalingstatisticsmustbeinvestigatedtofurtherunderstandandmodelhigh
resolutionrainfall.
• Theaforementionedpreviousworks[28,30,31]arebasedonavailablerainfalldata(S-POLradar,
TRMMsatelliteandraingauges),anditisdifficulttounderstanddifferencesoftheq-statistics
intemporalandspatialscalesduetofactorssuchastheintermittencyofrainfall,recordlength,
space-timeresolutionofdatasets,andgeographicsetting.
• Thenumberofbinsintheprobabilitymassfunctionconstitutesacentralissuetoquantifyentropic
measures. Previous studies have shown that the scaling exponent of Shannon entropy under
aggregationintimeitisnotsensitivetoeitherthenumberofbins [28]orrecordlength [30]. Then,
itisnecessarytostudythesensitivityoftheq-entropicmeasuresinordertochecktheirrobustness
tocharacterizing2-Dtropicalrainfallfields.
The objectives of this study are manifold. They involve questions based on the previous
studies [28,30,31],andnewonesregardingtheentropicscalingmeasuresofrainfall. Theobjectivesof
ourstudyarethus:
• TotesttheBL-Model[13]for2-DAmazonianrainfallfieldsconsideringtheEasterlyandWesterly
climaticregimes[55,56],andusingtheGeneralizedSpaceq-Entropy[31].
• Toexaminehowthespatialstructureofrainfallisreflectedintheq-entropicscalingmeasures
using the BL-Model and considering the influence of zeros in the GSEF through Montecarlo
experiments,aimedatunderstandingthesaturationoftheexponentΩ reportedby[28,30,31].
sat
• ToinvestigatetheconnectionbetweenparametersoftheBL-Model[13]andAmazonianrainfall
fieldsconsideringtheidentifiedclimaticregimes[55,56].
• Toquantifythesensitivityoftheq-entropicscalingstatisticstothenumberofbins andtothe
variabilityofrainfallintensity,inanattempttochecktherobustnessofsuchstatisticaltoolsinthe
multi-scalecharacterizationofrainfall.
• Tolinktwoimportanttheoreticalframeworks,namelystochasticprocesses(MultiplicativeCascades)
andInformationTheory(non-extensivestatisticalmechanics),toadvanceourunderstandingabout
thescalingpropertiesoftropicalrainfall.
Thepaperisorganizedasfollows. Section2describesthestudyregionanddataset. Section3
discusses the methods employed. Section 4 provides an in-depth discussion of results. Section 5
providesabriefdiscussionaboutthecriteriaandconditionstoestimateentropyingeophysicaldata.
Finally,Section6containstheconclusions.
2. StudyRegionandDataSets
GeneralInformation
Weuseasetof2-DradarrainfallfieldsgatheredinAmazoniaduringtheJanuary–February1999
WetSeasonAtmosphericMeso-scaleCampaign/LBA(WETAMC/LBA),whichwasdesignedtostudy
thedynamical,microphysical,electrical,anddiabaticheatingcharacteristicsoftropicalconvection
oversouthwesternAmazonia[19,59,63,64]. TheWETAMCcampaignwasdevelopedinthestateof
Rondônia(Brazil). Thedatasetusedinthepresentstudyconsistsonradarscansofstormintensities
recordedbytheS-POLradar(S-band,dualpolarimetric)locatedat61.9982◦W,11.2213◦S.Dataconsist
of2kmresolutionmicrowavebandreflectivitywhichisdirectlyrelatedtorainfallintensity,overacircle
of∼31,000km2. ScansproducedbytheColoradoStateUniversityRadarMeteorologyGroupwere
availableevery7–10minattheURLhttp://radarmet.atmos.colostate.edu/trmm-lba/rainlba.html.
Additionally, information about zonal wind velocity at 700 hPa during the study period was
obtainedfromtheNCEP/NCARReanalysis[65],overtheregioninside61◦ Wto62.8◦ Wand10.4◦ S
to12.1◦ S,correspondingtotheareacoveredbytheS-POLradar. Datawereobtained4timesperday
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at0000,0600,1200and1800LST.Inaddition,radiosondedata(62.37◦ W,10.75◦ S)wereusedfrom
the WETAMC campaign in Ouro Preto d’Oeste at the Fazenda Nossa Senhora site, located inside
theS-POLradarcoverageregion. RadiosondedatawereobtainedintheURLhttp://www.master.iag.
usp.br/lba/.
3. Methods
A set of experiments were developed to study the sensitivity of the GSEF of 2-D Amazonian
rainfallfieldsattemptingtolinkthespatialstructureofrainfallandtheemergingscalingexponents
oftheq-entropicanalysis[28,30]. TothataimweusetheBL-Modelproposedby[13]togenerate2-D
rainfallfieldsasamultiplicativerandomcascade,byvaryingthemodelparametersβandσ.Adetailed
descriptionofeachexperimentispresentedbelow.
3.1. ParametersoftheBL-ModelandAmazonianPrecipitationFeatures
Thefirstexperimentiscarriedoutbycontrollingthepercentageofwet(rainy)anddray(non-rainy)
areas,andthesecondonebycontrollingtheaverageintensityoftherainfallfield.Asetof1000simulations
foreachparameterwerecarriedout.Themassofrainfalloveratwo-dimensional(d =2)regionwas
consideredtheunit,thebranchingnumberb =4for2-Dcascades,andthelevelofsubdivisionn =6
(seeFigure1)duringallexperiments,consistentlywiththeobservedscansfromS-POLradarwhich
are64rows×64columnsmatrices(bn =46 =4096values). Ontheotherhand,withtheaimtolink
the numerical results from the BL-Model with the S-POL observations, we compared the samples
oftheestimatedcascade’sparameters(Equations(9)and(10))consideringβ vs. β and,
Easterly Westerly
σ vs. σ usingthek-sampletestsbasedonthelikelihoodratio[66],whicharemorerobust
Easterly Westerly
thantraditionalmethods(e.g.,Kolmogorov–Smirnov,Cramer–vonMises,andAnderson–Darling),
butalsobecausetheclimaticregimesprevailinginthestudyregionarestatisticallydifferentinterms
ofthecontinental-scaleflowandlightningactivity,aswellasintheverticalstructureofconvection
andotherprecipitationfeatures[56].
3.2. Bin-CountingMethodsandEntropicEstimators
The correct estimation of diverse informational entropy statistical parameters require to take
intoaccountdiversepracticalconsiderations. Gongandothers[67]arguethattherearefourpractical
problemsintheestimationofentropyusinghydrologicdata: (i)thezeroeffect;(ii)thewidelyused
bin-counting method for estimation of PDFs; (iii) the measure effect; and (iv) the skewness effect.
Wefocusourattentiononthesecondpracticalissuewithintheframeworkofscalingtheoryusing
Shannontheoreticalentropyinequality[68],
(cid:113)
S(T) ≤ln 2πeV(T), (17)
whereV(T) isthevarianceoftheprocessataggregationinterval T, withtheequalityholdingjust
forthefortheGaussiandistribution. Weuseasetofparametricandnon-parametricbin-counting
methods,suchasthoseintroducedbySturges[69],DixonandKronmal[70],Scott[71],Freedmanand
Diaconis[72],Knuth[73,74],ShimazakiandShinomoto[75,76],andarecentmethodforestimating
entropyinhydrologicdataproposedbyGongetal. [67]. Inwork,weusecommontechniquesreported
intheliterature,althoughthereareothermethods[77]. Finally,wediscusstheeffectofthenumberof
bins onthescalingofq-entropyinrainfallfields[28,30,31].
3.3. Sample-SizeandEntropyEstimators
Information-theory statistics require an adequate sample size for a proper estimation and
interpretation. In spite of the existence of a large body of literature dealing with the problem of
estimation of distributions for data sparsity and poor sampling [78,79], the problem constitutes a
challenge in geosciences. A recent study on the estimation of Shannon entropy in hydrological
Entropy 2017,19,685 9of25
recordsundersmall-samples[80],employedthreedifferentestimators: (i)maximumlikelihood(ML);
(ii)Chao–Shen(CS);and(iii)James–Stein-typeshrinkage(JSS).TheirresultsexhibitedthattheML
estimator had the worst performance of the three methods, with the largest Mean Squared Errors
(MSE) for all sample sizes. In particular, when sample sizes are small (less than 200 data points),
theentropyestimatorwasdramaticallyunderestimated,althougherrorsturnedouttodecreasequickly
whensamplesizesincreased. Furthermore,whenthesamplesizewaslargerthan100datapoints,
theaccuracyoftheCSandJSSestimatorswerebasicallythesame,withMSEnearlyequaltozero. Itis
worthmentioningthatthesaidstudy[80]didnotdealeitherwiththeeffectofthenumberofbins or
theroleofzerosintheestimationofentropy.
Attherootoftheproblemofthenumericalestimationofentropy,isthesample-sizenecessaryfor
adequatelyestimatingtheunderlyingprobabilitymassfunction(pmf)oftheprocess. Forexample,
it is well known that, for the ML estimators, the larger the sample size, the better the estimates
will be. In addition, high temporal resolution precipitation data sets (e.g., minutes or hours) are
mostly(approximately90%)constitutedbyzeros[30],whichitisnotthecaseforlow-resolutiondata
(e.g., months or years), so an adequate characterization of the tails of the Probability Distribution
Function(PDF)requireslongerdatasetsforhigh-resolutionrainfalldatathanforlow-resolution.
Withtheaimofstudyingtheinfluenceofsample-sizeintheGSEF,wewillcomparetheq-entropy
of the S-POL radar fields (matrices with 64 rows and 64 columns) and synthetic 2-D fields of the
BL-Modelatdifferentcascadelevels,n,toobtainfieldswithdifferentsizes,ξ,e.g.,thecascadelevel
n =1meansa2×2field,(2rowsand2columns)and,consequently,acascadeleveln =7meansa
128×128field. Inotherwords,afieldoftheBL-Modelwithn =1hasasample-sizeξ =4andafield
i
oftheBL-Modelwithn = 7hasasample-sizeξ = 16,384. Usingthesyntheticfields,wequantify
i
theq-entropy: (i)estimatingthepmfforallthevaluesinthesyntheticrainfallfield(includingzeros);
(ii) separing values in two subsets P(x) = {P(x =0),P(x >0)}; and (iii) the minimal quantity of
non-zerovaluesinthefields,toensurerobustnessintheestimationofS . Ourresultsarediscussed
q
inSection4.5.
3.4. Intermittencyandq-Order
Highspace-timeresolutiontropicalrainfallisahighlycomplexandintermittentprocess.Toanalyze
its intermittent behavior, several techniques have been proposed in the literature such as: spectral
scaleinvarianceanalysis[31,81–84];moment-scalinganalysis[1,3,18,19,31,32,84],andintermittency
exponents [84,85]. In this sense, the Generalized Space q-Entropy Function (GSEF) (Section1.4)
can be thought as a measure of (multi-) fractal behavior of a process [30,31]. In addition,
theBL-Model(Section1.2)isdirectlyrelatedtotheMandelbrot-Kahane-Peyriere(MKP)function[34,35]
characterizingthefractal(orscale-invariant)behaviorofamultiplicativecascadeprocess. Therefore,
aninterestingquestionarises:isthereanyrelationshipbetweentheGSEFandintermittencyestimators?
Inordertoshedlightaboutsuchquestion,theBL-Modelisusedtoestimatetheq-order,Θ ,andthe
q
multifractalitymeasure, M(q ,q ),definedbyBickel[86],Equation(22).
1 2
First, we carry out an experiment for high-resolution-spatial rainfall based on Bickel [86],
who showed the relationship between intermittency (multifractality) and the non-extensive order
(hereafterq-order)forpointprocessesofdimensionDrangingfrom0.1to0.9. Theorderinasystemis
definedintermsofitsdistancefromequilibrium,sothehigherdisordered,theclosertotheequilibrium
state. Theq-ordercanbedefinedas,
S
Θ(q) ≡1− q (18)
Smax
q
where S is the q-entropy (Equation (11)) and Smax is the maximum possible value of S for the
q q q
equilibriumcondition,whichprobabilisticallycanbedenotedas p =1/Nforallj =1,2,...,N,whose
j
Smax canbewrittenas,
q
1−N(1−q)
Smax = (19)
q q−1
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Atthispoint,itisnecessarytoemphasizethatEquation(19)isageneralizationfortheextensive
orderdefinedasameasureofcomplexityusingtheBoltzman-Gibbs-Shannonentropy[87]. Inaddition,
the non-extensive order satisfies 0 ≤ Θ(q) ≤ 1 with Θ(q) = 0 if P = 1/N ∀j and Θ(q) = 1 if
j
P = δ ∀j,givenanyintegerlbetween1andN[86,88].
j jl
Second,ageneralizationforΘ(q) ≡ Θ ,atmultiplespatialortemporalscales,ispossibleusing
q
asimilarproceduretoEquation(15). Therefore,ifthespatialscaleisdenotedasλ,andthemaximum
possiblevalueofq-entropyateachscaleλasSmax(λ),theq-ordercanberewrittenas,
q
S (λ)
Θ (λ) ≡1− q . (20)
q Smax(λ)
q
From previous studies [30,31], it is easy to note that Θ (λ) (cid:54)= S (λ) but for the scaling laws
q q
Θ (λ) ∼ λΩ(q) andS (λ) ∼ λΩ(q),thescalingexponentsΩ(q)areexactlythesame,whichmeansthat
q q
GSEF,Ω(q),doesnotchangeundersuchtransformation.
Third, considering that a process x(t) exhibiting a multifractal spectrum whose spectrum of
generalizedHurstexponents,H(q),isdefinedas[86],
ψ(q,T) = (cid:10)|x(t+T)−x(t)|q(cid:11)1/q ∝ TH(q), (21)
which holds for some range T with non-extensive parameter q. For ψ(q = 1,T) is the mean
(first moment) of the absolute displacement and for ψ(q = 2,T) is the standard deviation of this
displacement. Therefore,theintermittencyofanonstationaryprocess x(t)canbequantifiedbyits
multifractality, M(q ,q ),asthedifferencebetweentwogeneralizedHurstexponents,
1 2
(cid:40)−q q H(q2)−H(q1) q (cid:54)= q
M(q ,q ) = 1 2 q2−q1 1 2 (22)
1 2
limq1→q2 M(q1,q2) = −q22[∂H(q2)/∂q2] q1 = q2
normalized such that 0 ≤ M(q ,q ) ≤ 1 for nondegenerate processes, with M(q ,q ) = 0 for
1 2 1 2
monofractals[89].
From previously mentioned considerations, the generalized Hurst exponents for 2-D rainfall
fieldscanbecomputedassuggestedbyCarbone[90]incombinationwiththeEquation(21)asfollows,
 1/q
ψˆk(q) = (n −k)1(n −k) n∑x−kn∑y−k|x(i+k,j+k)−x(i,j)|q , (23)
x y i=1 j=1
wherex ∈ (cid:60)2,n andn arethenumberofrowsandcolumns,respectively. Theestimatesψˆ (q )and
x y k 1
ψˆ (q ) for k = k ,2k ,4k ,··· ,k , with k and k such that logψˆ (q ) and logψˆ (q )
k 2 min min min max min max k 1 k 2
exhibit linear relationship with logk. Then, Hˆ(q ) and Hˆ(q ) are the slopes of the least-square
1 2
regressions of logψˆ (q ) vs. logk and logψˆ (q ) vs. logk, respectively. Finally, the multifractality
k 1 k 2
isestimatedas,

−q q Hˆ(q2)−Hˆ(q1) q (cid:54)= q
Mˆ(q1,q2) = 1 2 q2−q1 1 2 (24)
limq1→q2 Mˆ(q1,q2) q1 = q2
In this work, we explore the relationship between intermittency (multifractality), Mˆ(q ,q ),
1 2
and q-order, Θ , for synthetic rainfall fields from the BL-Model. The results will be discussed
q
inSection4.6.