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Ecological
Modelling
jo u r n al ho m e p ag e :w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l
Estimating
humpback
whale
abundance
using
hierarchical
distance
sampling
Heloise
J.
Pavanato
a,∗,
Leonardo
L.
Wedekin
b,
Fernando
R.
Guilherme-Silveira
a,
Márcia
H.
Engel
b,
Paul
G.
Kinas
aaEnvironmentalStatisticsLaboratory,FederalUniversityofRioGrande,Km.8Itália,96201-900RioGrande,RS,Brazil
bInstitutoBaleiaJubarte/ProjetoBaleiaJubarte,26BarãodoRioBranco,Caravelas,BA,Brazil
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received25January2013
Receivedinrevisedform2May2017
Accepted8May2017 MSC: 13-52R1 Keywords: Megapteranovaeangliae Populationsize
Hierarchicallinetransectsampling
Bayesianinference
AtlanticOcean
a
b
s
t
r
a
c
t
WedevelopedaBayesiandistancesamplinganalysisusingahierarchicallystructuredmodel
param-eterizationtoestimatehumpbackwhaleabundanceintheSouthwestAtlanticOcean(BreedingStock
A).Weincludedcovariatesthataffectdetection(altitudeandsightingcue)andoccurrenceprobability
(yearanddistancefromshore).Populationsizesfor2008,2011and2015wereestimatedtobe7,689
(P.I.95%=6585–8931),8652(P.I.95%=7696–9682),and12,123(P.I.95%=10,811–13,531),respectively.
Theresultsindicateanaggregationofhumpbackwhalesinanintermediatedistancefromshoreline,an
increasingindensityfrom2008to2001andasubstantialoverlapbetweenposteriordistributionsof
densityfor2011and2015,whichsuggestsastabilizationofpopulationgrowthoverthelastyear.Our
parameterizationprovidedaclearviewofobservationalandecologicalprocessesandillustratesthatthe Bayesianhierarchicallinetransectapproachprovidesaflexibletooltoaccountforandevaluatevarious sourcesofuncertainty.
©2017ElsevierB.V.Allrightsreserved.
1. Introduction
Abundance and density are the most important demo-graphicmeasuresofapopulation(LiebholdandGurevitch,2002; MacKenzieetal.,2006)andareessentialinformationforpopulation assessments,viabilityanalyses,and evaluationsofmanagement procedures(Barlowetal.,1995;Carrettaetal.,2009;Wade,1998). Althoughimportant,onecommondifficultyinabundancestudiesis accountingfornegativebiasduetoundetectedindividualsbecause samplingproceduresdonotguaranteeaperfectdetectioninmost cases(Dorazioetal.,2006;RoyleandDorazio,2008).
Insomecases,itcanbereasonabletoconsidersampling variabil-ityandimperfectdetectiontobeconstantacrossspatialortemporal units.However,thisoptionisrarelyfeasible.Samplingprotocols andstatisticalinferenceshouldallowanexplicitandformal rep-resentationofdatausingconstituentmodelsofobservationsand underlyingecologicalorstateprocesses(e.g.MooreandBarlow, 2011;Royleetal.,2004;RoyleandDorazio,2008;Thomsonetal., 2008;Linkand Barker, 2010).Thestateprocessmodelsvary in
∗ Correspondingauthor.Presentaddress:DepartmentofMathematicsand
Statis-tics,UniversityofOtago,POBox56,Dunedin,Otago,NewZealand.
E-mailaddress:[email protected](H.J.Pavanato).
underlyingecologicalphenomena,whicharetheprimaryobject ofinference.Thisprocessismanifestinastatevariable,whichis typicallyunobservable.Incontrast,theobservationprocess con-tainsaprobabilisticdescriptionofmechanismsthatproducedata. Thisstructureisdescribedasahierarchicalorstate-spacemodel (RoyleandDorazio,2008).
The humpback whale (Megaptera novaeangliae) is a migra-toryspecies,movingseasonallybetweensummerfeedinggrounds at highlatitudes and winterbreeding groundsat low latitudes (Dawbin, 1956; Mackintosh, 1965). The International Whaling Commission(IWC)recognizessevenhumpbackwhalemigratory breedinggroundsintheSouthernHemisphere(IWC,1998).The populationofinterestinthisstudyisBreedingStockA,whichfeeds eastoftheScotiaSea,aroundSouthGeorgiaandtheSouth Sand-wichIslands,andbreedsintheSouthwestAtlanticOceanalongthe coastofBrazil(Engeletal.,2008;EngelandMartin,2009;Stevick etal.,2006;Zerbinietal.,2006).
Theearliestquantitativestudiesofhumpback whale popula-tionsuseddatafromcommercialcatches,whereitwaspossible toobtainrelativemeasuresof abundance(Clapham, 2000).The developmentofindividualidentificationtechniquesbasedonthe recognitionofventralflukepatterns(KatonaandWhitehead,1981) or,morerecently,genotyping(Palsbølletal.,1997)haveenabled theapplicationofmark-capturemethodstothisspecies(Hammond http://dx.doi.org/10.1016/j.ecolmodel.2017.05.003
Fig.1.Conceptualdiagramofthehierarchicallinetransectsamplingmodel
high-lightingthestateandobservationprocess.
et al., 1990).These methods are extremely usefulif individual recaptureprobabilitiesarehigh.However,theydonotperformas welliftheratesofrecapturearelow,whichcanbeaconsequence ofanimalsoccupyingrelativelylargeareas,asispresentlythecase withsomehumpbackwhalepopulations.Oneofthemostwidely usedtechniquesforquantifyingcetaceanabundanceislinetransect sampling.Thismethodissuggestedforwidelydistributed popula-tionsbecauseitislesssusceptibletofailuresofassumptionsthan aremark-recapturemethods(Bucklandetal.,2001,2004).
A long period of exploitation by whaling (which ended in 1972)causedlargepopulationdeclinesintheSouthernHemisphere humpbackwhalepopulations(Berzin,2008;Claphametal.,2005; Walsh,1999;Yablokov,1994).Themajorityofthesepopulations haveexhibitedsignsofrecoveryoverthepastdecades(Bannister andHedley,2001;Claphametal.,1999;Findlayetal.,1994,2011; Flórez-González,1991;Patersonetal.,1994;PatersonandPaterson, 1989),includingBreedingStockA(Wardetal.,2011).
ThefirstabundanceestimationforBreedingStockAwas con-ductedin1995usingmark-recapturemethods(Kinasetal.,1998). Sincethen,estimatesofabundancefromempiricaldataand assess-ment modelling have suggested that this population has been increasing(Andrioloetal.,2010;Freitasetal.,2004;Wardetal., 2011; Zerbiniet al.,2011).Someof these estimates(e.g. Ward etal.,2011)aresimilartothose obtainedthroughmodellingof knownlife-historyparametersfrompopulationsinvariousocean basins(e.g.mean rateof increaseof 7.3%year−1 for oneof the approachesusedbyZerbinietal.,2010).EvenifBreedingStockAis growingataratethatisapproachingitsmaximum,someconcerns remainbecauseofthehighuncertainty(evidencedthroughwide Bayesianprobabilityintervalsandfrequentistconfidenceintervals) associatedwiththeabundanceestimates,whichprecludeprecise estimatesofsomedemographicparameters,suchasthepopulation growthrate.
Giventheimportanceofcurrentabundanceestimatesandthe developmentoftechniquesthatincludehabitat-specific explana-toryvariablesoverdensity(e.g.HedleyandBuckland,2004),the goalofthisstudywastoestimatethesizeofBreedingStockAoff theBraziliancoastusinglinetransectsamplingwithinaBayesian statisticalframework. A hierarchicalmodellingapproach (Royle and Dorazio, 2008)was developed thatincorporates covariates thathavethepotentialtoaffectdetectionprobability(altitudeand sightingcue)andhumpbackwhaleoccurrence(yearanddistance fromshore)(Fig.1).Thedetectionprobabilityandabundanceof
Fig.2. Locationsofsightingsofhumpbackwhalegroups(red)andtransectlines(black)alongtheBrazilianbreedinggroundfor2008,2011and2015.Theisobathsof200m
humpbackwhales wereestimatedandcompared with frequen-tistestimates.Finally,asimulationstudywasusedtovalidateour approach.
2. Methods
2.1. Survey
Threesystematic sighting surveys wereperformed between AugustandSeptemberof2008,2011and2015,duringthepeakof humpbackwhaleabundanceinthebreedingseasonoffthe Brazil-iancoast(Martinsetal.,2001;Moreteetal.,2003).Ahigh-wing aircraft(Aerocommander) equipped withbubble windowswas usedtosurveythestudyareafromtheStateofRioGrandedoNorte (4◦34S)toRiodeJaneiro(23◦12S)in2008and2015.In2011,due tofundingconstraints,thesurveywasperformedfromtheState ofSergipe(10◦8S)toRiodeJaneiro(Fig.2).Paralleltransectlines weredesignedtosamplethesurveyareafromthecoastuntilthe 500misobath.Tomaximizethesamplingeffort,inthenorthern area(from Abrolhos Bank),transectsweredesigned ina zigzag shape due to thenarrow shelf. Aircraft flewduring favourable weatherconditionsatanaltitudeof1,000ft(304.80m)in2008 and500ft(152.40m)in2011and2015atanairspeedof110knots (56.59ms−1).
Observationswereconductedbytwoobserverslocatedatthe rightandleftbubblewindowsoftheaircraft.Foreachsighting,the declinationanglefromtheaircrafttothegroupofwhaleswas mea-suredbyahand-heldclinometerwheneveragrouppassedabeam. Thegeographicpositionofthesightingwasrecorded,alongwith thegroupsizeandsightingcuetype. Sightingcuewasgrouped intofourcategoriesthatconsidersimilarhumpbackbehaviours: (i)blow:animals blowing;(ii)aerialbehaviour: breaching; (iii) surface:flipperandtailslapping,lobtailingandthedorsalor ven-tralpartoftheanimal’sbodyonthesurface;and(iv)submerged: submergedanimals.
2.2. HierarchicalBayesianlinetransectmodel
Perpendiculardistances werecalculatedusing basic trigono-metricrulesfromtheaircraftaltitudeandthedeclinationangle tothesighting.Allanglesmeasuredbetween3◦and0◦(distances beyond3000m) werelater discardedbecause of measurement inaccuracy.
Wedevelopedahierarchicaldistancesamplingformulationby usingdataaugmentation(RoyleandDorazio,2008;LinkandBarker, 2010;KéryandRoyle,2016).Dataaugmentationisaconvenient schemeforobtainingBayesianposteriordistributionsviaMarkov chainMonteCarlomethods(TannerandWong,1987).Theimputed valuesfortheaugmenteddatainthisschemecanbeviewedasa specialwayofusingauxiliaryvariablestoaccelerateGibbs sam-plingalgorithms(LiuandWu,1999).
Letndenotethetotalnumberofgroupsforwhichxiarethe
knownperpendiculardistancesandyi=1istheobservedvariableof
detectionfori=1,2,...,n.Dataaugmentationconsistsofappending M−n‘observations’forsomelargevalueofM,withxi=NAdefined
asmissingvaluesandyi=0indicatingnodetectionfori=(n+1),...,
M.
ForeachoftheM‘observations’intheaugmenteddataset,the auxiliary indicatorvariablewi wasintroducedto representthe
group’soccurrence.Weassumethatwiareindependentrandom variableswithaBernoullidistribution,where istheprobability thatagroupfromthesuperpopulationMispartoftheeffective population.Itimmediatelyfollowsthatthetotalnumberofgroups Ng=
Mi=1wihasaBinomialdistributionNg∼Bin(M, ).Hence,inthisformulation,estimatingtheexpectedvalueofNgisequivalent
toestimating .Sincethevarianceofthebinomialdistributionis dependenton (var(Ng)=M (1− ))andgiventhatthebinomial
convergestothePoisson(M )asfarasthenumberoftrials(M) tendtoinfinityand tendsto0,wetesteddifferentMvaluestofit themodel.
Sincewejointlycomputedtheabundancein2008,2011and 2015,weassignedacategoricalcovariate‘year’onthelogitscale of toaddress differencesbetweenabundancein those years. Geographicregionhasalreadybeenusedina previousanalysis conductedintheBrazilianbreedinggroundthroughDistance soft-ware(Thomasetal.,2010),becausethesampledareahadbeen expanded,i.e.stratificationwasimplementedinthedesignphase (seeAndrioloetal.,2010fordetails).Inthepresentsurveys,the samplingswerecontinuous,andweoptedtointroduceahabitat covariatetoaccountforareasthathavepotentiallydifferent densi-ties.Thus,weincludeddistancefromshore(km)anditsquadratic termon toaccountforthisvariabilityasfollows:
log
ij 1+ ij =˛0[tij]+˛Zijfor j=1,2,3,where˛0 istheinterceptforeachyear(t=1,2,3for2008,2011,
2015),˛isthevectorofcoefficientsassociatedwiththecovariate matrixZ,whichcontainsdistancefromshore(d)anditsquadratic term(d2).
Anotherwaytouseasite-specificcovariatelikeyearondensity istodescribeabundance(N)asaPoissonrandomvariable,andthe intensityparameterasafunctionofcovariatesinadiscretespace model(KéryandRoyle,2016).Inthiscase,thedataaugmentation parameter isconfoundedwiththeinterceptofthePoissonmean andathinningprocessofthePoissonmeanisthenusedtospecify (seeKéryandroyle,2016,Chapter9forfurtherdetails).Becausewe aredealingwithacontinuousspacemodel,wepreferedtospecify thecovariatemodelinthedataaugmentationparameterinstead.
Toadddistancexasanindividualcovariatefori=1,2,...,M,we usedahalf-normaldetectionfunction:
g(xij)=exp
−x2 ij 22 ij .Weaddressedpossiblevariationsonthescaleparameterby extendingthedetectionfunctionto:
log(ij)=ˇ0[hij,cij],
whereˇ0isamatrixthatrepresentsthecoefficientofthe
inter-action term between altitude (h) and sighting cue (c). For the covariatessightingcueanddistancefromshore(d),M−ndatawere completedwith‘NA’andpriordistributionswereassumed: dij ∼ N(0,0.7)
cij ∼ Cat(K,p)
p ∼ Dir(K,a),
whereK=1,...,4(levelsofsightingcue)anda=a1,...,aK.
Since transect lines were randomly placed with respect to humpbackwhalegroups,distanceswereassumedtohavea uni-form distribution defined on some continuous interval (a, b), xij∼U(0,Xmax),whereXmaxisagivenmaximumatwhichdistance
measurementsweretruncated(3000minthisstudy).
Finally, we useda shifted Poisson distributionwith average group sizeby year (j+1) or equivalently (sij−1)∼Pois(j)to
changetheinferencefromgroups(Ng)toindividuals.
Ifwedefineajastheareasthatwereeffectivelycoveredbythe
surveys(i.e.areasinwhichgroupshadpositivedetection probabil-ity)andAjastheoverallstudyareainwhichthetransectlineshave
Table1
Priordistributionsassignedformodelparameters.
Parameter Prior ˛0 N(0,100) ˛1 N(0,100) ˛2 N(0,100) ˇ0 N(0,100) s U(0,10)
beenplaced,thentheparametersofprimaryinterestarepopulation
densityDjandsizeNj,whicharedefined:
Dj= M
i=1 wij sij aj Nj=Aj DjToaddresstheunrealisticassumptionofperfectdetectionat
distancex=0inaeriallinetransectsurveysofcetaceans(Buckland
etal., 2001), we includedg(0)asanuncertain,partiallyknown parameterbysimulatingfromN(0.67,0.15)(Andrioloetal.,2006) atthefinalstepoftheanalysis,aftertheestimateofNjhadbeen
obtained.TheIWCScientificCommitteehadrecommendedtheuse ofthisg(0)estimateasacorrectionfactorforaerialsurvey-based abundanceincorporatedin theassessment of BreedingStock A (IWC,2010).
WithintheBayesianframework,flatpriorswereassignedtoall parameters(Table1), andposteriordistributions wereobtained throughMarkovchainMonteCarlomethods(MCMC)byusingJAGS incombinationwithRsoftware(Kellner,2015).TheMCMC algo-rithmwasimplementedwith3chainsof400,000sampleseach, burn-inof 200,000andthinningof200,resultingin aposterior sampleof3,000values.Diagnosticstoverifyanyindicationsoflack ofconvergenceoftheMarkovchainswererunthroughthecoda package(Plummeretal.,2006).TheRcodeisavailableinAppendix A.
We compared our results with standard line transect esti-mates(Bucklandetal.,2001).Multiplecovariatedistancesampling (MCDS)(Bucklandetal.,2004;MarquesandBuckland,2003)was used to fit distance data by year along with the sighting cue covariateand a stratified estimate.We used thedefault analy-sis,whichestimatesvariance empiricallyandusesthesize-bias regressionmethodtoestimategroupsize.Wealsoselecteda mul-tipliertoaccountfor g(0)asrecommended byIWC (2010). We thenexaminedthefrequentistpointestimateincomparisonwith theBayesianposterior mean, frequentist confidence interval in comparisonwiththeBayesianprobabilityinterval,andcoefficient ofvariation (CV)ofboth frequentistandBayesiananalyses.We definedaCVfortheBayesiananalysisusingtheposteriormeanand standarddeviationofN.Hence,BayesianCVshaveadifferent mean-ingastheydonotrelatetothefrequentistsamplingdistributionof estimates.
2.3. Simulationstudy
Weperformedasmallsimulationstudytoevaluatetheprecision ofhumpbackwhaleparametersestimatedthroughthehierarchical linetransectmodel.
Thisstudywasperformedfortwoyearswithdifferentsurvey altitudes(denotedby1and2)byestablishingasuperpopulation M=1,500.Numericcovariateson weresimulatedthroughN(0, 1)(suchasthestandardizedcovariatedistancefromshoreinthe humpbackwhaledataset), whereasfactor covariatesonwere simulatedasacategoricaldistributionwiththreelevelsand
dif-Table2
Posteriorsummary(mean,medianandstandarddeviation)ofparameters˛0,˛1and
˛2regardingtheinfluenceofthecovariatesyear,distancefromshoreandquadratic
termonoccurrenceprobability( );ˇ0[h,c]regardingtheinfluenceofthe
interac-tiontermbetweenaltitude(h=1:1000ft,h=2:500ft)andsightingcue(c=1:blow,
c=2:aerialbehaviour,c=3:submerged,c=4:surface)onscaleparameter();and
sregardingmeangroupsize.
Parameter Mean Median SD
˛0[2008] −2.528 −2.528 0.088 ˛0[2011] −2.228 −2.229 0.068 ˛0[2015] −1.999 −1.999 0.065 ˛1 0.006 0.006 0.031 ˛2 −0.167 −0.166 0.022 ˇ0[1,1] −0.504 −0.519 0.129 ˇ0[1,2] −0.515 −0.670 1.28 ˇ0[1,3] −1.563 −1.568 0.127 ˇ0[1,4] −1.005 −1.007 0.073 ˇ0[2,1] −0.413 −0.420 0.091 ˇ0[2,2] 7.647 6.763 5.176 ˇ0[2,3] −1.781 −1.783 0.087 ˇ0[2,4] −0.871 −0.872 0.047 s[2008] 1.600 1.599 0.045 s[2011] 1.557 1.556 0.037 s[2015] 1.587 1.585 0.034
ferentreplacementprobabilitiesof0.5,0.3and0.2.Perpendicular
distanceswerealsosimulatedfromU(0,1).
Theparameters˛0[1]=0.5,˛0[2]=0.7,˛1=0,and˛2=−0.3and
ˇ0[1,1]=−0.5,ˇ0[1,2]=−1, ˇ0[1,3]=−1.5,ˇ0[2,1]=−0.7,ˇ0[2,
2]=−1.2andˇ0[2,3]=−1.6werefixedtoderivethelatent
parame-ters andthroughlinearregressionsonthelogitandlogarithmic
scales,respectively.
After adjusting the perpendicular distances using the
half-normalfunction,weobtainedthedetectedgroupsy1andy2asa
binomialtrialofthetruepopulationsizesofgroupsNg1=837and
Ng2=897and thedetectionprobabilities.Note thatthenumber
ofdetectedgroups(n1=444andn2=394)isobtainedas
Mi=1yi.
Groupsizesweresimulated througha shiftedPoisson
distribu-tionwithmeans1=s2=1.5.Wethenobtainedthepopulationsize
of individuals as Ni=
Ngi=1si which resulted in N1=1250 and
N2=1366.
Theparameterswereestimatedusingthesamelinetransect
modeldevelopedabove.DifferentvaluesforMwerealsotested
untiltheNgvariancereachedstabilization.Finally,theestimated
parameters were compared with the true parameter values in
termsofthemeanand95%probabilityintervalofposterior
dis-tributions.
3. Results
3.1. Humpbackwhaleabundance
Atotalof325,445and607groupsofhumpbackwhaleswere
observedin2008,2011and2015,respectively,inaninferencearea
of130,546.2km2for2011and174,154.9km2for2008and2015.
Thegroupsizesrangedfrom1to8.WeusedasuperpopulationM
withanupperboundof9,000groups(withoutincorporatingA/a
tosavecomputationtime),whichwasconsideredtoexceedany
realisticnumber ofgroupswithintheeffectivelycoveredareaa
andenoughtoprovideaPoissonapproximation.
Aposteriorsummaryoftheparametersrelatedtooccurrence
probabilityand perpendiculardistancesis presentedinTable2.
There was a significant effect of distance from shore and its quadratic term (posterior of parameters did not cover zero), whereintheoccurrenceprobabilitypeakswerefoundin72,86and 65kmfor2008,2011and2015,respectively(Fig.3).Theoccurrence probability significantlyincreasedacrosstheyears(Table2).
0 50 100 150 200
0.03
0.05
0.07
distance from shore (km)
ψ2008 0 50 100 150 200 0.03 0.05 0.07 0.09
distance from shore (km)
ψ2011 0 50 100 150 0.06 0.08 0.10 0.12
distance from shore (km)
ψ2015
Fig.3.Occurrenceprobabilities asafunctionofdistancefromshorefor2008,
2011and2015.Thinredlinesindicatethedensitypeakrelatedtodistancefrom
shore.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereader
isreferredtothewebversionofthisarticle.)
0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.8 altitude = 1, blow perpendicular distance (x) g(x) 0.2 0.4 0.6 0.8 0.4 0.8
altitude = 1, aerial behaviour
perpendicular distance (x) g(x) 0.0 0.1 0.2 0.3 0.4 0.2 0.8 altitude = 1, submerged perpendicular distance (x) g(x) 0.0 0.2 0.4 0.6 0.8 0.2 0.8 altitude = 1, surface perpendicular distance (x) g(x) 0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.7 altitude = 2, blow perpendicular distance (x) g(x) 0.0 0.2 0.4 0.6 0.8 1.0 0.9999994
altitude = 2, aerial behaviour
perpendicular distance (x) g(x) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.6 altitude = 2, submerged perpendicular distance (x) g(x) 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0 .8 altitude = 2, surface perpendicular distance g(x)
Fig.4.Half-normal detectionfunctionsfittedto perpendicular distanceswith
effectsofaltitudeandsightingcueeffect(blow,submerged,surfaceandaerial
behaviour).Blacklinesindicatetheposteriormedian;thingreylinesindicatethe95%
probabilityinterval.Altitude=1correspondsto500ft(2011and2015);altitude=2
correspondsto1,000ft(2008).
Regardingthehalf-normalfunctionfittedtodistances,forthe altitudeof1,000ft(2008),allsightingcuesproducedadecreasein (Table2).Forthealtitudeof500ft(2011and2015),thesame patternwasobservedexceptforthesightingcueaerialbehaviour, whichhadastrongpositivemeaneffectonbutwithalarge asso-ciateduncertainty(Table2andFig.4).Themeangroupsizesranged from1.558to1.601andwerenotdifferentacrossyears(Table2).
Thepopulationsizeestimateshadanincreasingpatternfrom 2008 to 2015 (Fig. 5), as did the density estimates (Table 3). Consideringthesmallestareasurveyed(2011),themeanannual rate of increase was 5.21% (SD=0.747), which represents an
N2008 Frequency 5000 7000 9000 11000 0 40000 80000 1 20000 N2011 Frequency 7000 8000 9000 11000 0 40000 8 0000 140000 N2015 Frequency 9000 11000 13000 15000 0e+00 4 e+04 8e+04
Fig.5.Humpbackwhaleabundanceestimatesfor2008,2011and2015.Histograms
representthefullposteriordistributions;thinredlinesindicatetheposteriormean;
thingreylinesindicatethe95%probabilityinterval.(Forinterpretationofthe
ref-erencestocolourinthisfigurelegend,thereaderisreferredtothewebversionof
thisarticle.)
Table3
Summaryofposteriordistributionofthehierarchicallinetransectmodelbyyear:
mean,coefficientofvariation(%)and95%probabilityinterval.Nisabundance;Dis
density.
Year N D CV P.I.95% Area(km2)
2008 7689 0.044 7.78 6585–8931 174,154.9
2011 8652 0.066 5.87 7696–9682 130,546.2
2015 12,123 0.07 5.71 10,811–13,531 174,154.9
Table4
Summaryofparameters(pointestimate,coefficientofvariationand95%confidence interval)estimatedthroughthemulticovariatedistancesamplingusingsighting cueeffectandpost-stratification.E(s)isexpectedgroupsize;Nisabundance;Dis density.
Year Parameter Pointestimate CV(%) 95%C.I.
2008 E(s) 1.766 2.82 1.670–1.867 N 10,286 28.26 5770–18,335 D 0.064 28.26 0.036–0.115 2011 E(s) 1.653 2.19 1.583–1.725 N 13,782 26.71 8026–23,664 D 0.106 26.71 0.062–0.182 2015 E(s) 1.699 2.00 1.634–1.767 N 16,690 24.94 9983–27,901 D 0.104 24.94 0.0624–0.174
increaseof475individualsbyyear(SD=83).However,theannual
growthfrom 2008 to2011 was largerthan that from 2011to
2015(r2008−2011=11.08%;r2011−2015=1.173%).Analysingthe
den-sityposteriordistributions, there isalmostnooverlap between
densitiesin2011and2008,whereasthereisasignificantoverlap
betweendensitiesin2011and2015(Fig.6).
TheresultsobtainedfromMCDSaresummarizedinTable4.The standardabundanceestimateswerehigherthanthoseestimated bytheBayesianhierarchicalmodelwithadifferenceof2,597for 2008,5,130for2011and4,567individualsfor 2015(calculated asthedifferencebetweenthemaximumlikelihoodandposterior meanestimates).TheCVfortheBayesianinferencewasonaverage 76%smallerthanthestandardinference.TheBayesian95% prob-abilityintervalsfor2008and2015arecompletelycoveredbythe frequentistconfidenceinterval(between2.5%quantileandpoint
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0 20 40 60 80 100 120 D poste rior dist ri b ution 2008 2011 2015
Fig.6. Humpbackwhaledensityestimatesfor2008,2011and2015.Densityplots
representthefullposteriordistributions.
0.07 0.08 0.09 0.10 0.11 0.12 0 20 50 ψ1 poste rior dist ri b ution 0.08 0.09 0.10 0.11 0.12 0 20 40 60 ψ2 poste rior dist ri b ution
Fig.7.Occurrenceprobabilityparameters( 1and 2)estimatedforthe
simula-tionstudy.Redlinesindicatethetrueparametervalue;thinblacklinesindicatethe
posteriormean;thingreylinesrepresentthe95%probabilityinterval.(For
inter-pretationofthereferencestocolourinthisfigurelegend,thereaderisreferredto
thewebversionofthisarticle.)
estimate).For2011,thelowerquantileoftheprobabilityinterval isoutsidethefrequentistconfidenceintervalbyasmallfraction.
3.2. Simulationstudy
Thetrue parametervalues werealwayscovered by the95% probabilityintervalofposteriordistributions(Figs.7–9).Notethat, sinceweincreasedthesuperpopulationsizeMtoassurereliable varianceestimates,thevectorofparamaters˛werenolonger cor-respondenttotheparamatersfixedinthesimulationexperiment. Forthisreason,weshowtheposteriordistributionsof ,which werereadilyobtainedthrough =Ng/M,insteadof˛.
Themeanposteriordistributionswerequitesimilartothetrue valuesforallparameters(Figs.7–9),althoughtherewasatendency tooverestimatethetrueparametervaluesfor˛0(Fig.7),ˇ0[2,1],
−0.8 −0.6 −0.4 −0.2 024 β0[1,1] poste rior dist ri b ution −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 024 β0[2,1] poste rior dist ri b ution −1.2 −1.1 −1.0 −0.9 −0.8 −0.7 −0.6 024 β0[1,2] poste rior dist ri b ution −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1.0 024 β0[2,2] poste rior dist ri b ution −2.0 −1.8 −1.6 −1.4 −1.2 01234 β0[1,3] poste rior dist ri b ution −1.8 −1.6 −1.4 −1.2 01234 β0[2,3] poste rior dist ri b ution
Fig.8. Parametersregardingdetectionprobability(ˇ0[1,1],ˇ0[2,1],ˇ0[1,2],ˇ0[2,
2],ˇ0[3,1]andˇ0[3,2]).Redlinesindicatethetrueparametervalue;thinblacklines
indicatetheposteriormean;thingreylinesrepresentthe95%probabilityinterval.
(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderis
referredtothewebversionofthisarticle.)
0.40 0.50 0.60 02468 10 s1 poste rior dist ri b ution 0.40 0.50 0.60 0 2 468 10 s2 poste rior dist ri b ution 1000 1200 1400 0.000 0.002 0.004 N1 poste rior dist ri b ution 1100 1300 1500 0.000 0.002 0.004 N2 poste rior dist ri b ution
Fig.9. Parametersregardinggroupsize(s1ands2)andabundance(N1andN2)
esti-matedforthesimulationstudy.Redlinesindicatethetrueparametervalue;thin
blacklinesindicatetheposteriormean;thingreylinesrepresentthe95%
probabil-ityinterval.(Forinterpretationofthereferencestocolourinthisfigurelegend,the
readerisreferredtothewebversionofthisarticle.)
ˇ0[1,2]andˇ0[2,3](Fig.8)andunderestimateforˇ0[2,2],ˇ0[1,
3](Fig.8)andN2(Fig.9).
4. Discussion
4.1. Modellingissues
We developed an extensionof thehierarchical linetransect modelproposedbyRoyleetal.(2004),inwhichaBayesian for-mulationusingdataaugmentationwasgivenbyRoyleandDorazio (2008)andLinkandBarker(2010).First,weextendedthe analy-sisinatemporalscale(threeyears),whichallowedtheestimation ofabundanceacrossyearsbypoolingdistancedata.Second,we insertedcovariateswithpotentialtoadjustforoccurrence proba-bilityatthelevelofobservationalunits.
Toimprove precision and reducethe bias of estimates,line transectsamplingsaregenerallyanalysedusingstratification, com-putingabundanceindependently by geographicregions (strata) (Bucklandetal.,2001).Andrioloetal.(2010)definedfiveblocks (i.e.strata)atthedesignstepoftheaerialsurveysfrom2002to 2004,and this wasfurtherexpanded to eight blocks for 2005. Theabundancesineachblockwerecomputedindependentlyand pooledtoobtaintotalpopulationsizes.From2008onward,weno longerperformeddesignstratificationintheaerialsurveys.Rather, thetransectswerecontinuouslytravelleddependingonweather andlogisticalconditions.Thus,post-stratification(i.e.stratification ofdataaftercollection) isapossibilitysinceencounterratecan differinspace(Wedekinetal.,2008).Post-stratificationis straight-forwardtoimplementina hierarchicalsetting byintroducinga categoricalindividualcovariatethatassignseachdetection(group) toastratum(seeKéryandRoyle,2016chapter9foranexample).
Includinghabitat-specificvariablesindistancesamplingdata isofinterestinmanystudies.Forexample,HedleyandBuckland (2004)fittedspatialmodels(linearandadditivemodels)toline transectdata.ABayesianapproachforspatialinferenceon den-sityfromlinetransectsurveydatawasdevelopedbyNiemiand Fernández(2010), in which density was described as an inho-mogeneousPoissonprocesswhoseintensityfunctionincorporates covariateeffectsandspatialsmoothingofresidualvariation.Inboth studiesandinthis work,theassumptionoftransectsrandomly placedinrelationtogroups(orindividuals)canberelaxedsincewe wereabletolinkcovariateeffectstovariationondensity.Thus,this approachmightbeusedwithdataarisingfromsourcesotherthan designed-basedsurveys,forinstance,opportunisticdata.The hier-archicalBayesianlinetransectmodelcouldbeextendeddepending onthepurposeofthestudy.Forexample,itispossibletoinsert otherdetectionandecologicalcovariateson andby specify-ingpriordistributiononcovariates;usingotherdetectionfunctions (Oedekovenetal.,2014);andaddingdualobserverdatatoestimate perceptionbiasrelatedtodetectiononthetrackline(Connetal., 2012;Guilherme-SilveiraandKinas,2016).Althoughthesetypes ofmodelsareconceptuallystraightforwardtoimplementandto interpret,theirdisadvantageisthehighdemandforcomputational timeduetothedataaugmentationscheme,whichcanhinder anal-ysesoflargedatasetswithmanycovariates.Thismaybeovercome byusinga reversiblejumpalgorithm (rjMCMC),whichisuseful whenthedimensionalityoftheparameterspaceisunknown(Link andBarker,2010;Connetal.,2012;Oedekovenetal.,2014).
Oursimulationstudyprovidesevidencethat thehierarchical modelisaccurate.Theestimated95%probabilityintervalsalways includedthetrueparametervalues,butdeviationsfromthe poste-riormeantothetrueparametervalueswereidentified.However, wedonotbelievethattheserelativelysmalldifferencesarea prob-lemthat mightpreventanyfurtherinference.However,a more completesimulationstudyisadvisedinhelpingtoclarifyinwhich situationstheBayesianhierarchicallinetransectmodelusingdata augmentationmightfail.
Eventhoughweconsideredg(0)<1regardingavailabilitybias in our analysis, the population size estimated here may be biased downwards. A preliminary analysis indicates that the average group size is underestimated by approximately 17.5% (P.I.95%=13.9–21.1%)inaerialsurveyswhencomparedtovessel surveys,suggestingthatonceacorrectionfactorforthisproblemis included,theestimatedabundanceshouldincrease.Another neg-ativebiasthatwearenottakingintoaccountisperceptionbias, whichcorrespondstothefractionofgroupsmissedbyobservers foraseriesofreasons,includingfatigue,poorsightingconditions, inexperience,andsoforth(MarshandSinclair,1989).Inhigh den-sityareas,theperceptionbiasmightincreaseasmanyavailable groupscanbemissedduetothesmalltimeframeresultingfrom theaircraftspeed.Concernedwiththistypeofbias,weare
cur-Fig.10.AbundanceestimateswithrespectiveintervalsbasedonCVsfrom1995to
2015.Darkgreysquaresindicateestimatesderivedfrommark-recaptureanalyses
(Kinasetal.,1998;Freitasetal.,2004);lightgreytrianglesindicateestimatesderived
fromstandardaeriallinetransectsamplinganalyses(Andrioloetal.,2006,2010);
yellowtrianglesindicateestimatesderivedfromstandardvessellinetransect
sam-plinganalyses(Bortolottoetal.,2016);redtrianglesindicateestimatesderivedfrom
thehierarchicallinetransectsamplingmodel.EstimatefromZerbinietal.(2004)
werenotincludedduetoitssmallfractionofBreedingStockAsurveyed.(For
inter-pretationofthereferencestocolourinthisfigurelegend,thereaderisreferredto
thewebversionofthisarticle.)
rentlydevelopinghierarchicallinetransectmodelsusingadouble observerconfiguration.Thesetwo biasesmightexplain the dif-ferencebetweenaerialandvesselsurveysaswaspointedoutby Bortolottoetal.(2016).Forinstance,in2008afrequentist anal-ysisusingshipsurveydataestimated16,410whales(CV=0.228) (Bortolotto et al., 2016)while the corresponding aerial survey frequentistestimationwas10,286(CV=0.283)andtheBayesian estimation was7,689(CV=0.078).On theotherhand,although biasregardingdoublecountingislikelysmallinlinetransect sam-pling,itbecomesapotentialproblemifanimalsoftenaredouble countedinthesamesamplingunit.Itmayleadtooverestimationof encounterratesandabundance.Observersonboardavesselatan averagespeedof17kmh−1(Bortolottoetal.,2016)havethe poten-tialtocountthesamegroupmorethanonceinhighdensityareas, andaevaluationofthedimensionofthisbiasshouldbeatopicof futureresearch.
4.2. Ecologicalimplications
TheresultsofAndrioloetal.(2010)pointedtoblockD, corre-spondingtothecentreoftheAbrolhosBank,ashavingthehighest densityfrom2002to2005.Ourresultindicatesthatdensitypeaks occur in intermediate distances from shore (65–86km), which agreeswithAndrioloetal.(2010)andMartinsetal.(2001).Our studyproposedtheintroductionofahabitatcovariateintotheline transectsamplinganalysis,whichleadstoaclearpictureofhow humpbackwhalesusetheirbreedinghabitatontheBraziliancoast. Thisapproachisalsoimportantforevaluatingwhetherandhow concentrationareaschangeacrossyears.Thisnotonlyimproves ourunderstandingoftheecologyofthespeciesbutshouldalso leadtomoreeffectiveconservationmeasures.
Thepresentresults,obtainedthreeyearsafterthelast avail-ableestimatefromaerialplatform(Andrioloetal.,2010),indicated thatthepopulationconsistentlygrewuntil2011(Fig.10).However,
whenanalysingtheposteriordistributionsestimateforhumpback density(Fig.6),theoverlapbetweendensitiesin2011and2015 suggestsadecreaseintherateofgrowthandmightbeevidence thatthepopulationismovingtowardsstabilization.Toelucidate whetherthe populationis stabilizing and reachingits carrying capacity,prospectivesurveysshouldbecarriedoutandaproper evaluationshouldbedrawn.
Acknowledgements
We thank the observers Bruno B. Pitarello, Inês L. Serrano, IraêT.G. Oliveira,Kátia R. Groch, Mariana C. Nevesand Milton C.C.Marcondesfortheirhelpduringtheaerialsurveys.Wealso thank allpilots for safely conductingus during theflights. We aregratefulforthevaluablecomments ofAlexandreN.Zerbini, EduardoR.SecchiandLucianoDallaRosaonaearlierversionof themanuscript. We are also gratefulfor the revision provided byJ.AndrewRoyleandSamanthaStrindbergwhichsignificantly improvedthis manuscript.PaulConnand LenThomasprovided insightfulsuggestionsregardingthemodel.Petrobrasistheofficial sponsorofProjetoBaleiaJubarte/InstitutoBaleiaJubarte.Thiswork wassupported byPetrobras, FUNBIO,CMA/ICMBio and Veracel CeluloseS.A.TheConselhoNacionaldeDesenvolvimento Cientí-ficoeTecnológico(CNPq)providedascholarshiptothefirstauthor, whoworkedundertheguidanceofthelast.
AppendixA.
ThecodebelowimplementsthehierarchicalBayesianline tran-sectmodelusedtoestimatehumpbackwhaleabundancethrough JAGS.Thevariablesareasfollows:
•x:perpendiculardistancesmeasuredtoeachanimalorgroupdetected
•gs:observedgroupsize
•t(year),d(distancefromshore),h(altitude)andc(cue):covariates
#ModelinJAGSsyntax
model{
#Likelihood:
for(iin1:M){M=superpopulationsize
for(jin1:J){J=years
w[i,j]∼dbern(psi[i,j])
logit(psi[i,j])<-alfa0[t[i,j]]+alfa1*d[i,j]+alfa2*d[i,j]*d[i,j]
d[i,j]∼dnorm(0,0.7)
x[i,j]∼dunif(0,1)
np[i,j]<--((x[i,j]*x[i,j])/(2*sigma2[i,j]))
sigma2[i,j]<-sigma[i,j]*sigma[i,j]
log(sigma[i,j])<-beta0[h[i,j]*c[i,j]]
c[i,j]∼dcat(x[])
p[i,j]<-exp(np[i,j])
mu[i,j]<-w[i,j]*p[i,j]
y[i,j]∼dbern(mu[i,j])
gs[i,j]∼dpois(g[j])
wgs[i,j]<-w[i,j]*(gs[i,j]+1)
}j }i N2008<-sum(wgs[1:M,1]) N2011<-sum(wgs[1:M,2]) N2015<-sum(wgs[1:M,3]) } # References
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