Ageneralframeworkforanimaldensityestimationfromacousticdetectionsacrossaﬁxedmicrophonearray
BenC.Stevenson
1,
*,DavidL.Borchers
1
,ResAltwegg
2
,Ren
eJ.Swift
3
,DouglasM.Gillespie
3
andG.JohnMeasey
4
1
CentreforResearchintoEcologicalandEnvironmentalModelling,UniversityofStAndrews,TheObservatory,BuchananGardens,StAndrews,FifeKY169LZ,UK;
2
StatisticsinEcology,EnvironmentandConservation,DepartmentofStatistical Sciences,andAfricanClimateandDevelopmentInitiative,UniversityofCapeTown,Rondebosch7701,SouthAfrica;
3
SeaMammalResearchUnit,ScottishOceansInstitute,UniversityofStAndrews,StAndrews,FifeKY168LB,UK;and
4
Centrefor InvasionBiology,DepartmentofBotanyandZoology,StellenboschUniversity,Stellenbosch,SouthAfrica
Summary
1.
Acoustic monitoring can be an eﬃcient, cheap, noninvasive alternative to physical trapping of individuals.Spatially explicit capture
–
recapture (SECR) methods have been proposed to estimate calling animal abundanceanddensityfromdatacollectedbyaﬁxedarrayofmicrophones.However,thesemethodsmakesomeassumptionsthatareunlikelytoholdinmanysituations,andtheconsequencesofviolatingtheseareyettobeinvestigated.
2.
WegeneralizeexistingacousticSECRmethodology,enablingthesemethodstobeusedinamuchwidervariety of situations. We incorporate timeofarrival (TOA) data collected by the microphone array, increasing theprecision of calling animal density estimates. We use our method to estimate calling male density of the CapePeninsulaMossFrog
Arthroleptellalightfooti
.
3.
Our method gives rise to an estimator of calling animal density that has negligible bias, and 95% conﬁdenceintervals with appropriate coverage. We show that using TOA information can substantially improve estimateprecision.
4.
Our analysis of the
A. lightfooti
data provides the ﬁrst statistically rigorous estimate of calling male densityforananuranpopulationusingamicrophonearray.Thismethodﬁllsamethodologicalgapinthemonitoringof frogpopulationsandisapplicabletoacousticmonitoringofotherspeciesthatcallorvocalize.
Keywords:
anura, bootstrap, frog advertisement call, maximum likelihood, Pyxicephalidae, spatiallyexplicitcapture
–
recapture,timeofarrival
Introduction
Population size is one of the most important variables in ecology and a critical factor for conservation decisionmaking.Distance sampling and capture
–
recapture are both wellestablished methods used for the estimation of animal abundanceand density. Both approaches calculate estimates of detectionprobabilities, and these provide information about how manyanimalsinthesurveyareawereundetected.Estimatesofabundance and density are then straightforward to calculate. Oneparticular point of diﬀerence is that distance sampling useslocations of detected individuals in
space
, while typically capture
–
recapture records the initial capture, and subsequentrecaptures, of individuals at various points in
time
. The relatively recent introduction of spatially explicit capture
–
recapture (SECR) methods (Eﬀord 2004; Borchers & Eﬀord 2008;Royle &Young2008; Royle
et al.
2013; see Borchers2012; fora nontechnical overview) has married the spatial componentof distance sampling and the temporal nature of capture
–
recapture approaches. Indeed, Borchers
et al.
(in press) linkedthetwounderaunifyingmodeltoshowthattheyexistatopposite ends of a spectrum of methods, which vary with theamountofspatialinformationemployed.Data collected from SECR surveys are records (known asthe capture histories) of where and when each individual wasdetected. Detection may occur in a variety of ways, for example, by physical capture, or from visual recognition of a particular individual. SECR methods treat animal activity centres asunobserved latentvariables,and the positionsofdetectors thatdid (and did not) detect a particular individual are informativeabout its location; an individual’s activity centre is likely to beclosetothedetectorsatwhichitwasdetected.Eﬀord, Dawson & Borchers (2009) ﬁrst proposed the application of SECR methods to detection data collected withoutphysically capturing the animals themselves, but from anacoustic survey using an array of microphones (see section 9.4,Royle
et al.
2013; for a summary of acoustic SECR methods).This is appealing when the species of interest is visually crypticand diﬃcult to trap physically, but is acoustically detectable.Moreover, it is less disruptive and invasive than physical capture. When individuals can be detected (virtually) simulta
*Correspondenceauthor.Email:bcs5@standrews.ac.uk
©
2014 The Authors. Methods in Ecology and Evolution
©
2014 British Ecological Society
MethodsinEcologyandEvolution
2015,
6,
38–48 doi: 10.1111/2041210X.12291
neously on multiple detectors (e.g. by virtue of the same callbeing recordedatmultiplemicrophones),then ‘recaptures’(or,more accurately, ‘redetections’) occur at diﬀerent points inspace rather than across time, thus removing the need for multiple survey occasions. This has the advantage of substantiallyreducing the cost of ﬁeldwork. In this case, the capture histories simply indicate which microphones detected each call, andnolongerhaveatemporalcomponent.Thelatentlocationsareno longer considered activity centres, but simply the physicallocation of the individual when the call was made. The use of SECR for these data is advantageous over competingapproaches (e.g. distance sampling) as these often assume thatthe locations can be determined without error, and this doesnotholdinmanycases.The method of Eﬀord, Dawson & Borchers (2009) used signal strengths (i.e. the loudness of a received call at a microphone) to improve estimates of individuals’ locations:microphones that received a stronger signal of a particular callare likely to be closer to the latent source locations than thosethat received a weaker signal. Such additional information iscapable of improving the precision of parameter estimates(Borchers
et al.
inpress).Naturally, acoustic detection methods are unable to estimate the density of noncalling individuals. Any density estimates obtained from acoustic surveys therefore correspond tothe density of calling individuals, or density of calls themselves(i.e. callsper unit area per unit time), rather than overall population density. If the proportion of individuals in the population that call is known (or can be estimated), then it isstraightforward to convert estimated calling animal density topopulation density. Otherwise, the utility of measures relatedto abundance or density (e.g. relative abundance indices) hasbeen shown for a variety of taxa, of which only subsets of thepopulationsareacousticallydetectable.For example, females do not call for almost all anuran species.Itisthereforeonlypossibletoobtainanestimateofcallingmaledensityfromanacousticsurvey.Nevertheless,qualitativeestimates of call density (i.e. density recorded on a categoricalscale) for frog populations have been found to correlate wellwith capture
–
recapture estimates (Grafe & Meuche 2005), andmale chorus participation is the best known determinant of mating success in many frog species (Halliday & Tejedo 1995).Asaresult,calldensity isoften usedasa proxyforfrogdensity(e.g. Corn, Muths & Iko 2000; Crouch & Paton 2002; Pellet,
Helfer&Yannic2007).Further examples of taxa for which measures related toabundance and density have been estimated using acousticmethods include birds (e.g. Buckland 2006; CelisMurillo,Deppe & Allen 2009; Dawson & Eﬀord 2009), cetaceans (e.g.Harris
et al.
2013; Martin
et al.
2013), insects (e.g. Fischer
et al.
1997) and primates (e.g. Phoonjampa
et al.
2011). SeeMarques
et al.
(2013) for an overview of the use of passiveacousticsfortheestimationofpopulationdensity.While the method of Eﬀord, Dawson & Borchers (2009)shows promise in estimating calling animal abundance anddensity using ﬁxed arrays of acoustic detectors, a major practical issue was not addressed in this work: the method asdescribed is only appropriate if each individual is only detectable on a single occasion (e.g. by virtue of making exactly onecall).The likelihoodpresentedassumesindependentdetectionsbetween calls, thus independence between call locations. Thisis unlikely to hold when individuals emit more than a singlecall, as locations of calls made by the same individual arealmost certainly related. This issue was not explicitly acknowledged, and as a result, the subsequent analyses presented byMarques
et al.
(2012) and Martin
et al.
(2013), which applythe method of Eﬀord, Dawson & Borchers (2009), are problematic. Additionally, the analysis of Dawson & Eﬀord (2009)used an approach that is unlikely to be appropriate in manyscenarios.Weoutlinethesestudiesbelow.Marques
et al.
(2012) and Martin
et al.
(2013) appliedacoustic SECR methods to data collected by underwater hydrophones, which detected vocalizations from minke whales
Balaenoptera acutorostrata
Lac
ep
ede. As the location of awhale’s call is likely to be close to the location of its previouscall, this analysis suﬀers the assumption violation mentionedabove.Theconsequencesofthisviolationarenotclear.Furthermore, calls were treated as the unit of detectionmeaning that each call (rather than each individual) was givenits own capture history. The resulting density estimate wastherefore of call density rather than calling whale density. Distance sampling analyses have previously used independentlyestimated call rates to convert from call density to calling animal density (e.g. Buckland 2006), and Eﬀord, Dawson & Borchers (2009) suggest using the same approach. The eﬃcacy of this approach in an SECR setting is yet to be investigated, andawayofestimatingvarianceofanimaldensityestimatesgeneratedinthiswayhasnotyetbeenproposed.Dawson & Eﬀord (2009) estimated density of singing ovenbirds
Seiurus aurocapilla
(Linnaeus) using small arrays of microphones. Of all calls attributed to the same individual,only the ﬁrst was retained for analysis. Assuming independence between locations of retained calls was therefore appropriate, and the resulting density estimate was of singing birddensity. However, there are two potential problems with thispractice: ﬁrst, it can only be carried out in situations whereindividuals are recognizable from their calls, and on many surveys, this is not the case. Second, recall that the likelihoodassumeseachindividualisonlydetectableonasingleoccasion.Therefore, any detections retained for analysis must be detections of the ﬁrst call the individual made
over the course of thesurvey
, and not only the ﬁrst call that was
detected
. In general,it isnot known when a call is undetected, and so onecannotbesure that the ﬁrst detected call is the ﬁrst call. Retaining callsthat were the ﬁrst detected call, but not the ﬁrst emitted call,canresultinpositivebiasincallinganimaldensityestimates.Putting the method of Eﬀord, Dawson & Borchers (2009)into practice is therefore problematic. It is necessary to investigate the consequences of violating assumptions of call locationindependence and propose suitable estimators based on acoustic detection data from a microphone array.Inthis manuscriptwe present a general method that gives rise to estimators of calling animal density. We also develop methodology that canbe used to estimate variance of the proposed estimators. We
©
2014 The Authors. Methods in Ecology and Evolution
©
2014 British Ecological Society,
Methods in Ecology and Evolution
,
6
, 38–48
Animal density estimation from a ﬁxed microphone array
39
show by simulation that both perform well under reasonableassumptions.Anadditionalimprovementispossible,whichwealsoincorporate into our estimator. While Eﬀord, Dawson & Borchers(2009) suggest the use of received signal strengths to furtherinform call locations (in addition to detection locations), Borchers
et al.
(in press) demonstrate the utility of timeofarrivalinformation in this regard. Multichannel arrays are capable of recordingtheprecisetimesatwhichasignalisdetectedbyeachindividual microphone, and subtle diﬀerences between thesetimes are informative about the location of the sound source.For example, a call’s source location is likely to be closest tothe microphone with the earliest detection time. The use of such auxiliary data informative on call locations in acousticSECR is further motivated by Fewster & Jupp (2013), whoshowthatincorporatingresponsedatafromadditionalsourcesleads to estimators that are asymptotically more eﬃcient.Indeed,weshowviasimulationthatourestimatorhaslessbiasandismoreprecisewhenitincorporatestimeofarrivaldata.We use our method to estimate calling male density of theCape Peninsula Moss Frog
Arthroleptella lightfooti
(Boulenger) from an acoustic survey. The genus
Arthroleptella
(mossfrogs; family Pyxicephalidae) are tiny (adults are typically 7
–
8 mm total length), visually cryptic and inhabit seepages onmountain tops in South Africa’s Western Cape Province(Channing 2004). Due to the region’s topography, many species are severely range restricted, endemic to individual mountains, such that most of the genus are on the IUCN red list (1Critically Endangered, 1 Vulnerable, 3 Near Threatened and 2LeastConcern;Measey2011).Individuals are extremely hard to ﬁnd (approximately 3
–
4personhours per individual) and therefore prohibitivelyexpensive to monitor via direct observation. However, malescan be heardcalling throughout the austral winter from withinmontane seepages, making an acoustic survey ideal. Movement of individuals is minimal over the courseof such surveys;during physical searches, frogs appear to call from the samepreciselocations(Measey,pers. obs.).Currently,these populations are monitored with a subjective estimate of calling maleabundance (Measey
et al.
2011). Such subjective methods aretypicallyemployedinanuranmonitoringmethodologies(Dorcas
et al.
2009). These estimates have no corresponding measure of estimate uncertainty. Additionally, there is no formalway of accurately determining the survey area within whichindividuals are detected, and so estimates of calling male density are not available. Indeed, Dorcas
et al.
(2009) concludethat current auditory monitoring approaches to surveyinganuran populations are restricted in their ability to estimateabundance or density. At present, no method exists that iscapable of generating both point and interval estimates of either call or calling male density in a statistically rigorousmanner. For the genus
Arthroleptella
(among others), thisproblem is further compounded by the lack of any methodcapable of identifying individuals from their calls, so it is notknown how many diﬀerent individuals have been detected.Themethodwepresentovercomestheseproblems.
Materialsandmethods
OVERVIEW
Ourmethodhasthreemaincomponents:
1.
AnacousticSECRsurveyfromwhichcalldensityisestimated.
2.
Estimation of the average call rate, allowing for conversion of thecalldensityestimateintoacallinganimaldensityestimate.
3.
Aparametricbootstrapprocedureforvarianceestimation.Once call density is estimated in Step 1, establishing an estimate forthe mean call rate in Step 2 allows for the estimation of calling animaldensity. Measures of parameter uncertainty (such as standard errorsand conﬁdence intervals) are calculated using a parametric bootstrapapproach. Parameter estimates from both Step 1 and Step 2 arerequiredinordertocarryoutthisprocedure.The SECR model we present for Step 1 assumes that individual callsareidentiﬁable,thatis,itisknownwhetherornottwodetectionsatdifferent microphones are of the same call. Some acoustic preprocessingis required in order to ascertain how many unique calls were detectedacrossthearray andwhich of these were detectedbyeachof the microphones. The details of this process will vary from study to studydepending factors such as the acoustic properties of the focal species’calls. We later describe a simple method used for the application to
A.lightfooti
,whichissuitableforoursurvey.We do not assume that individuals are identiﬁable, that is, ourmethod does not require knowledge of whether or not two detectedcallsweremadebythesameanimal.Thisismorediﬃcultthanidentifying calls; there is less information available from which to determineindividual identiﬁcation, and one must contend with betweencall variationinwhateveracousticpropertiesofthecallsaremeasured.
NOTATION AND TERMINOLOGY
We consider a survey of duration
T
with
k
microphones placed atknown locations within the survey region
A
R
2
. Vocalizations frommembers of the focal species are detected by these microphones, andmeasurementsofthereceivedsignalstrengthandtimeofarrivalarecollectedforeachdetection.Adetectionisdeﬁnedtobeareceivedacousticsignal of a call that has a strength above a particular threshold,
c
, sothat is easily identiﬁable above any background noise. Detections withstrengthsbelowthisthresholdarediscarded.The observed data comprise the number of unique calls detected,
n
c
,capture histories of the detected calls,
Ω
, recorded signal strengths,
Y
,and times of arrival measured from some reference point (typically thebeginningofthesurvey),
Z
.Thesearedeﬁnedasfollows.Let
x
ij
be 1 if call
i
2 f
1
;
. . .
;
n
c
g
was detected at microphone
j
2
{
1,
. . .
,
k
}
, and 0 otherwise. We denote
x
i
=
(
x
i
1
,
. . .
,
x
ik
) as the capture history for the
i
th
call on the
k
detectors, and
Ω
contains the capture histories for all
n
c
calls. If the
i
th
call was detected by the
j
th
microphone, then we also observe
y
ij
and
z
ij
, the measured signalstrength and the recorded time of arrival from the start of the survey,respectively. The sets of all these observations are given by
Y
and
Z
,and
y
i
and
z
i
contain the signal strength and timeofarrival information associated with the
i
th
call.The detected calls have unobserved locations
X
¼ ð
x
1
;
. . .
;
x
n
c
Þ
,where
x
i
2
A
provides the Cartesian coordinates of the location atwhichthe
i
th
callwasmade.Wealsouse
x
genericallytodenoteaparticularlocationwithinthesurveyregion.Notethatlocationsofcallsemitted by the same individual cannot be considered independent. As it is
©
2014 The Authors. Methods in Ecology and Evolution
©
2014 British Ecological Society,
Methods in Ecology and Evolution
,
6
, 38–48
40
B. C. Stevenson
et al.
notknownwhichcallsweremadebythesameindividual,calllocationsingeneralarenotindependent.The parameter vector
h
=
(
D
c
,
c
,
/
) is estimated from the acoustic survey data. The scalar
D
c
is call density (calls per unit area per unit time),which is assumed to be constant across the survey area covered by thearray (although see the discussion for comments on modelling spatialvariation in calling animal and call density), while the vectors
c
and
/
containparametersassociatedwiththesignalstrengthandtimeofarrivalprocesses,respectively.The detection function and the eﬀective sampling area (ESA) playimportant roles in both SECR and distance sampling, and so they areworth brieﬂy introducing here. The detection function
g
(
d
;
c
) gives theprobability that a call is detected by a microphone, given that theirrespectivelocationsareseparatedbydistance
d
.Thisisusuallyamonotonicdecreasingfunctionascallsfurtherfromamicrophoneareusuallyless detectable. Here, we use the signal strength detection function (Efford, Dawson & Borchers 2009; further detail provided in below), andthis depends on the signal strength parameters
c
. Assuming independence across microphones, the probability that a call made at
x
isdetected at all is therefore
p
ð
x
;
c
Þ¼
1
Q
k j
¼
1
1
g
ð
d
j
ð
x
Þ
;
c
Þ
, where
d
j
(
x
) is the distance between the location
x
and the
j
th
microphone. TheESAdependsonthedetectionfunctionandisgivenby
a
(
c
)
=
∫
A
p
(
x
;
c
)
d
x
(Borchers&Eﬀord2008;Borchers2012).
Theaveragecallrateofcallingmembersofthepopulationatthetimeofthesurvey,
l
r
,isestimatedfromaseparate,independentsampleof
n
r
callrates,
r
¼ ð
r
1
;
;
r
n
r
Þ
.If
r
isusedtoestimateaparametricdistribution for population call rates, then the vector
w
holds the associatedparameters.Theﬁnalparameterofinterestiscallinganimaldensity,
D
a
.Throughout this manuscript, we do not explicitly diﬀerentiatebetween a random variable and its observed value, instead this shouldbe clear from its context. Likewise, we use the function
f
(
) to generically denote any probability density function (PDF) or probabilitymassfunction(PMF)withoutexplicitdiﬀerentiation.Therandomvariable(s) that
f
(
) is associated with should be clear from its argument(s).From Equation (2) onwards, we omit the indexing of parameters inPDFsandPMFsforclarity.
CALL DENSITY ESTIMATOR
The estimator we propose for
h
is based on an SECR model, which wedescribeinthissection.The full likelihood is the joint density of the data collected from theacousticsurvey,asafunctionofthemodelparameters:
L
ð
h
Þ ¼
f
ð
n
c
;
X
;
Y
;
Z
;
h
Þ¼
f
ð
n
c
;
D
c
;
c
Þ
f
ð
X
;
Y
;
Z
j
n
c
;
c
;
/
Þ
:
eqn 1Note that
D
c
does not appear in the second term of Equation (1).This is a consequence of assuming that call density is constant over thesurveyarea(Borchers&Eﬀord2008).SECRapproachesoftenassumethatthenumberofanimalsdetectedisaPoissonrandomvariable,asanimallocationsareconsideredarealization of a Poisson point process. Because we do not know how manyunique individuals have been detected, the distribution of the randomvariable
n
c
is not known (indeed, it is certainly not a Poisson randomvariableifindividualscallmorethanonce,seeAppendixS3).Thisissueis linked to the dependence of withinanimal call locations; independence in call locations implies that said locations are a realization of aPoissonpointprocess,butanydependenceviolatesthis.We use the socalled
conditional likelihood
approach of Borchers &Eﬀord (2008), which we extend here to include signal strength andtimeofarrival information. This allows for estimation of
h
withoutany distributional assumption on
n
c
, by conditioning on
n
c
itself.Parameters
c
and
/
can be estimated directly using this likelihood,whichisthesecondterminEquation (1):
L
n
ð
c
;
/
Þ¼
f
ð
X
;
Y
;
Z
j
n
c
Þ
:
eqn 2Oncetheestimate
b
c
hasbeenobtained,anestimateof
D
c
canthenbecalculated using a Horvitz
–
Thompsonlike estimator. This is accomplished by dividing the number of detected calls by the estimated ESAandthesurveylength,thatis
b
D
c
¼
n
c
a
ð
b
c
Þ
T
:
eqn 3Estimates for SECR model parameters that are obtained viamaximization of the full likelihood are in fact equal to thoseobtained via maximization of the conditional likelihood and use of a Horvitz
–
Thompsonlike estimator (Borchers & Eﬀord 2008), sothere is no practical diﬀerence in the two approaches if we are onlyinterested in point estimates (though note that this only holds whendensity is assumed constant across the survey area). Indeed, specifying the distribution for the number of detections (here denoted as
n
c
) only serves to allow calculation of estimate uncertainty; here,
b
D
c
depends on
n
c
, and so uncertainty in
b
D
c
is subject to the variance of
n
c
.Let us now describe the conditional likelihood, Equation (2), in further detail. The capture histories,
Ω
, received signal strengths,
Y
, andtimes of arrival,
Z
,all depend on thecall locations
X
: the closer acall ismade to a microphone, the higher the probability of detection, the louder the expected received signal strength, and the earlier the expectedmeasured time of arrival. We therefore obtain the joint density of
Ω
,
Y
and
Z
,conditionalon
n
c
,bymarginalizingover
X
:
L
n
ð
c
;
/
Þ ¼
Z
A
nc
f
ð
X
;
X
;
Y
;
Z
j
n
c
Þ
d
X
¼
Z
A
nc
f
ð
X
;
Y
;
Z
j
X
;
n
c
Þ
f
ð
X
j
n
c
Þ
d
X
¼
Z
A
nc
f
ð
Y
;
Z
j
X
;
X
;
n
c
Þ
f
ð
X
j
X
;
n
c
Þ
f
ð
X
j
n
c
Þ
d
X
:
By assuming independence between the detected calls’ recorded signal strengths and times of arrival, conditional on
X
(i.e. the time of acall’sdetectiondoesnotdependonitsstrength),weobtain
L
n
ð
c
;
/
Þ¼
Z
A
nc
f
ð
Y
j
X
;
X
;
n
c
Þ
f
ð
Z
j
X
;
X
;
n
c
Þ
f
ð
X
j
X
;
n
c
Þ
f
ð
X
j
n
c
Þ
d
X
:
The conditional likelihood presented above is intractable for tworeasons: (i) in general, the joint density of the call locations,
f
ð
X
j
n
c
Þ
, isunknown as we are unable to allocate calls to individuals – the dependence between call locations is not known and (ii) the integral is of dimension 2
n
c
, usually rendering any method of its approximation toocomputationallyexpensivetobefeasible.Instead, we compute the
simpliﬁed likelihood
that overcomes thesetwoproblemsbytreatingcalllocationsasiftheyareindependent.Justiﬁcationforthisisthattreatingnonindependentdataasiftheyareindependent often has minimal eﬀect on the bias of an estimator (thoughvariance estimates may be aﬀected substantially). This gives
f
ð
X
j
n
c
Þ¼
Q
n
c
i
¼
1
f
ð
x
i
Þ
and results in a separable integral, allowing forthe evaluation of a product of
n
c
2dimensional integrals instead of asingle2
n
c
dimensionalintegral:
L
s
ð
c
;
/
Þ¼
Y
n
c
i
¼
1
Z
A
f
ð
y
i
j
x
i
;
x
i
Þ
f
ð
z
i
j
x
i
;
x
i
Þ
f
ð
x
i
j
x
i
Þ
f
ð
x
i
Þ
d
x
i
:
eqn 4
©
2014 The Authors. Methods in Ecology and Evolution
©
2014 British Ecological Society,
Methods in Ecology and Evolution
,
6
, 38–48
Animal density estimation from a ﬁxed microphone array
41
Estimatesfor
c
and
/
are found by maximizing the log of the simpliﬁedlikelihoodfunction,thatis
ð
b
c
;
b
/
Þ¼
arg
max
c
;
/
log
L
s
ð
c
;
/
Þð Þ
;
eqn 5andourestimatorfor
D
c
remainsasshowninEquation (3).In situations where call locations
can
be considered independent, theconditional and simpliﬁed likelihoods are equivalent. Otherwise, thesimpliﬁed likelihood is not a true likelihood per se and should not betreated as such. That is, any further likelihoodbased inference (such asthe calculation of standard errors based on the curvature of the loglikelihood surface at the maximum likelihood estimate, or likelihoodbasedinformationcriteria)shouldnotbedirectlyused.The following sections focus on providing further details about eachtermthatappearsintheintegrandofEquation (4).
Signalstrength
The use of signal strength to improve estimator precision in SECRmodelswasﬁrstproposedbyEﬀord,Dawson&Borchers(2009).Assuming independence between received signal strengths (see thediscussion forcomments on thispoint),theﬁrst component of the integrandinEquation (4)is
f
ð
y
i
j
x
i
;
x
i
Þ¼
Y
k j
¼
1
f
ð
y
ij
j
x
ij
;
x
i
Þ
:
The expected received signal strength of the
i
th
call at the
j
th
microphone can be any sensible monotonic decreasing function of
d
j
(
x
i
), thedistance between the
j
th
microphone and the location of the
i
th
call.Here,wesimplyuse
E
ð
y
ij
j
x
i
Þ¼
h
1
ð
b
0
s
b
1
s
d
j
ð
x
i
ÞÞ
;
where
h
1
(
) is the inverse of a link function (typically chosen to beeither the identity or log function). See Dawson & Eﬀord (2009) foralternative speciﬁcations of the expected signal strength. We accountforGaussianmeasurementerrorinthereceivedsignalstrengths,thatis
y
ij
j
x
i
N
ð
E
ð
y
ij
j
x
i
Þ
;
r
s
Þ
:
The parameter vector
c
therefore comprises
b
0
s
,
b
1
s
and
r
s
that havedirect signalstrengthinterpretations:
b
0
s
isthesourcesignalstrength of calls (on the link function’s scale),
b
1
s
is the loss of strength per metretravelled due to signal propagation (on the link function’s scale), and
r
s
is the standard deviation of the normal distribution used to accountforsignalmeasurementerror.However,recallthat
y
ij
isonlyobservedifthereceivedsignalstrengthexceeds the microphone threshold of detection, that is, if and only if
y
ij
>
c
(or, equivalently,
x
ij
=
1). Otherwise,
y
ij
is discarded and
x
ij
is setto 0. Therefore, we set
f
(
y
ij

x
ij
=
0,
x
i
) to 1, and (
y
ij

x
ij
=
1,
x
i
) is a randomvariablefromatruncatednormaldistribution,giving
f
ð
y
ij
j
x
ij
¼
1
;
x
i
Þ¼
1
r
s
f
n
y
ij
E
ð
y
ij
j
x
i
Þ
r
s
1
U
c
E
ð
y
ij
j
x
i
Þ
r
s
1
;
eqn 6where
f
n
(
)and
Φ
(
)arethePDFandthecumulativedensityfunctionof thestandardnormaldistribution,respectively.
Probabilityofdetection
Based on the previous section, Eﬀord, Dawson & Borchers (2009) proposed the
signal strength detection function
, to be used when signalstrength information has been collected by the detectors during anSECRsurvey.Thistakestheform
g
ð
d
;
c
Þ¼
1
U
c
h
1
ð
b
0
s
b
1
s
d
Þ
r
s
;
thus giving the probability of a call’s received signal strength exceeding
c
(and,therefore,theprobabilityofdetection).The
i
th
capture history,
x
i
, is only observed if the
i
th
call is detected,that is if
P
k j
¼
1
x
ij
[
0. Thus, we observe
x
i
conditional on detection,and so
f
(
x
i

x
i
) must incorporate the probability of detection in thedenominator. Assuming independent detections of each call across allmicrophones, the third component of the integrand in Equation (4) istherefore
f
ð
x
i
j
x
i
Þ¼
Q
k j
¼
1
f
ð
x
ij
j
x
i
Þ
p
ð
x
i
;
c
Þ
:
As
x
ij
is1ifthe
i
th
callisdetectedbythe
j
th
microphone,and0otherwise,wehave
f
ð
x
ij
j
x
i
Þ¼
g
ð
d
j
ð
x
i
Þ
c
Þ
x
ij
¼
1
;
1
g
ð
d
j
ð
x
i
Þ
c
Þ
x
ij
¼
0
:
eqn 7
Timeofarrival
Asingledetectiontime on itsownis not informative on calllocation.Itis only
diﬀerences
between precise arrival times that provide information about the relative position of a call in relation to the locations of the microphones at which it was detected. Timeofarrival data aretherefore only informative for calls detected at two or more microphones; the arrival times,
z
i
, depend on
x
i
through
m
i
, the number of microphones that detected the
i
th
call, that is
m
i
¼
P
k j
¼
1
x
ij
,
m
i
2
{
1,
⋯
,
k
}
.Therefore,
f
(
z
i

x
i
,
x
i
)
f
(
z
i

m
i
,
x
i
),andweset
f
(
z
i

m
i
=
1,
x
i
)to1.Information about call locations improves the precision of parameter estimates, though here we do not assume that times of arrival allowperfect triangulation of call locations. Instead, we account for uncertainty in recorded times of arrival due to Gaussian measurement error,controlled by the parameter
r
t
. For calls detected at two or moremicrophones,inferencecanbemadebymarginalizingoverthetimethecall was made, a latent variable, and this integral is available in closedform (see the online supplementary material of Borchers
et al.
inpress),
f
ð
z
i
j
m
i
[
1
;
x
i
Þ¼ ð
2
pr
2
t
Þ
ð
1
m
i
Þ
=
2
2
T
ﬃﬃﬃﬃﬃ
m
i
p
exp
X
f
j
:
x
ij
¼
1
g
ð
d
ij
ð
x
i
Þ
d
i
Þ
2
2
r
2
t
0@1A
:
eqn 8The term
d
ij
(
x
i
) is the expected call production time, given call location
x
i
, and the time of arrival collected by detector
j
, that is
d
ij
(
x
i
)
=
z
ij
d
j
(
x
i
)/
v
,where
v
isthespeedofsound.Theaverageacrossalldetectorsonwhichadetectionwasmadeis
d
i
Calllocations
We assume individuals’ locations are a realization of a homogeneousPoisson point process across the survey area,
A
. As the dependencebetweencalllocationsisnotclear,it isnot possibletospecifytheirjointdensity,
f
(
X
), from data collected by the acoustic survey alone. Underthe simpliﬁed likelihood (Equation 4),this isnowtractable:
X
itself isarealizationofaﬁlteredhomogeneousPoissonpointprocess
–
theintensity of
emitted
calls is constant across the survey area, but the intensityof
detected
calls is highest closest to the microphones. The ﬁltering is
©
2014 The Authors. Methods in Ecology and Evolution
©
2014 British Ecological Society,
Methods in Ecology and Evolution
,
6
, 38–48
42
B. C. Stevenson
et al.