ENDANGERED SPECIES RESEARCHEndang Species Res

Vol. 4: 113–122, 2008doi: 10.3354/esr00068

Printed January 2008Published online January 18, 2007

INTRODUCTION

It is notoriously difficult to estimate the spatial distrib-ution of marine mammals. Marine mammals at sea arehard to discern, and species determination is often diffi-cult, even for trained observers. However, the situationhas changed considerably with the introduction ofsatellite-linked telemetry. Satellite telemetry has inher-ent limitations, such as low accuracy in positioning ofmarine mammals. Marine mammals are most oftentracked by the Argos satellite system (e.g. Stewart et al.1989, Teilmann et al. 1999, Laidre et al. 2002). Argospositions have variable accuracy, and the distance fromthe calculated to the true position ranges from a fewhundred metres to more than 50 km (Vincent et al.2002, White & Sjöberg 2002). These positioning errorscan affect conclusions to various degrees, dependingon the behaviour of the species tagged and the specific

questions addressed. In studies of off-shore species thatmigrate 100s to 1000s of km, a low accuracy may not becritical to determining spatial distribution and area use.In contrast, determination of the distribution and areause of species that move over short distances withincomplex habitats is almost always limited by the accu-racy in positioning. Area use is commonly described byestimating the home range of the animal. Home rangecan be expressed in various ways, such as fixed kernels(e.g. Worton 1989, Seaman & Powell 1996, Born et al.2004) or minimum convex polygons (Mohr 1947). Othermodels of spatial use also take advantage of a prioriknowledge of the behaviour of the species, obtained byother means (e.g. VHF-telemetry and aerial counts athaulout sites; Matthiopoulos 2003).

This study presents a new strategy, originally devel-oped for compilation of data for the atlas of Danishmammals (Baagøe & Jensen 2007) but with consider-

© Inter-Research 2008 · www.int-res.com*Email: jat@dmu.dk

THEME SECTION

Harbour seal spatial distribution estimated fromArgos satellite telemetry: overcoming positioning

errors

Jakob Tougaard1,*, Jonas Teilmann1, Svend Tougaard2

1National Environmental Research Institute, University of Aarhus, Department of Arctic Environment,Frederiksborgvej 399, PO Box 358, 4000 Roskilde, Denmark

2Fisheries and Maritime Museum, Tarphagevej 2, 6710 Esbjerg V, Denmark

ABSTRACT: A limiting factor of satellite telemetry in the context of habitat use by marine mammalsis the low accuracy of the received positions. A novel statistical analysis to overcome the low accuracywas developed in the context of processing data on harbour seals Phoca vitulina for the atlas of Dan-ish mammals. Ten harbour seals were caught in the Danish Wadden Sea and tracked with satellitetransmitters. The statistical analysis reversed the problem of positioning: Instead of attempting to cor-rectly assign each individual position to a single grid cell, our approach considers the combined prob-ability that at least one position originated in each grid cell. Thus, all satellite-derived positions,including positions of poor precision, can contribute to the evaluation. The method is an alternativeto other methods describing spatial use, such as kernel home range, and constitutes a viableapproach for inclusion of satellite-derived positional data into spatial modeling of animal distributionand habitat use.

KEY WORDS: Service Argos · Positioning accuracy · Grid survey · Abundance · Kernel home range ·Phoca vitulina · North Sea · Spatial modelling

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OPENPEN ACCESSCCESS

Endang Species Res 4: 113–122, 2008

able potential in the context of spatial modelling ofspecies’ distribution. The method uses available infor-mation on the precision of all positions, including thoseof low precision, obtained by analysing all positions asa whole together with a measure of their individualprecision. This makes it possible to identify those gridcells most likely to have been visited by at least 1 ani-mal. The distribution range achieved is thus conserva-tive, in the sense that only areas with high confidenceof animal presence are included.

MATERIALS AND METHODS

Tagging of animals. Ten harbour seals Phoca vitulina(5 males, 5 females) of different age classes (Table 1)were equipped with Argos satellite transmitters (SDR-T16 with 4 flat M1 batteries, Wildlife Computers). Ani-mals were caught on Rømø Island in theDanish Wadden Sea on 3 separate occa-sions in the first half of 2002 and trans-mitters were glued to the fur on the topof the head with fast hardening epoxyglue (Araldite 2012). Transmitterstransmitted daily with a maximum of500 uplinks allowed per day and wereprevented from transmitting during thenight hours (22:00 to 03:00 h local time)when satellite coverage was poor. Me-dian transmission time was 77 d (range49 to 100 d) and a median of 3.8 posi-tions per day per transmitter was ac-quired (range 1.5 to 4.6 d–1). Thirteenpercent of the positions belonged to Lo-cation classes 1 to 3 (most precise); theremaining were 0, A and B (least pre-cise). See Tougaard et al. (2003) for ad-ditional information on capture and tag-ging. Capture and tagging of animalswas carried out with permission fromthe Danish Ministry of Environment,Forest and Nature Agency.

Use of positions in the atlas of Dan-ish mammals. The atlas of Danishmammals (Baagøe & Jensen 2007)compiles all recent verified observa-tions of mammals on Danish territory(including the exclusive economiczones, EEZ), separated into 10 × 10 kmUniversal Transverse Mercator (UTM)grid cells. The fundamental logic of theatlas is simple: only grid cells with veri-fied observations of a particular speciesare to be included as positive, and thuspresence but not density is assigned to

the grid cells. As there are very few verified observa-tions of seals from the open sea, it was suggested thatthe available satellite telemetry data be included, eventhough the precision of most of the Argos positions wasof the order of several km. The precision of most obser-vations was not sufficient to assign them with certaintyto a particular 10 × 10 km grid cell. The challenge thenwas to rephrase the presence/absence question of thegrid survey in statistical terms and identify cells as pos-itive for presence when the likelihood that at least 1seal is present is above a predetermined criterion. Oneexample is shown in Fig. 1, which illustrates a section of15 UTM grid cells with positions received from 10harbour seals. The question is: Which cells are to becounted as positive for presence (at least 1 observationof a seal) and which are not? Cells to the NW are clearlyempty and cells LH40, LH51 and LH61 are likely to bepositive, but the status of the others is less clear.

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No. Sex Weight Length Date tagged No. of trans- No. of (kg) (cm) (yy/mm/dd) mission days positions

1 M 33 141 02/01/04 84 2552 F 24 107 02/01/04 75 3443 F 26 111 02/01/04 81 3054 M 25 116 02/01/04 69 2275 M 110 172 02/02/18 92 3366 F 43 143 02/02/18 78 3557 M 110 164 02/02/18 100 1548 F 44 134 02/05/06 50 1149 M 47 139 02/05/06 62 28010 F 52 150 02/05/06 49 208

Table 1. Phoca vitulina. Details of the tagged harbour seals. Positions trans-mitted by Argos satellite transmitter

Easting (km)320 330 340 350 360 370

6200

6210

6220

6230

1

2

B

AB

B

B

A

B

A

B

B

B

B

B

1

B

B

1

1BB

A

101 A

A

B

B

B

0

A

B

B

B

B

B

BA

B

B

0

LH22

LH21

LH20

LH32

LH31

LH30

LH42

LH41

LH40

LH52

LH51

LH50

LH62

LH61

LH60

Nor

thin

g (k

m)

Fig. 1. Example of Argos positions received from 3 harbour seals in the NorthSea. Shown are 15 Universal Transverse Mercator squares (each 10 × 10 km)from the central part of the Danish North Sea. Received positions of the taggedharbour seals are indicated with letters or numbers corresponding to their

precision (Location class, see Table 2)

Tougaard et al.: Argos telemetry and harbour seal abundance

The simplest solution to the problem is to includeonly positions with high precision (Location class 1 to3; 68% error percentile 1000 m or less, see ‘Model’below). However, a substantial amount of informationis lost this way (87% of all positions in the presentstudy would be discarded); neither does this solve theproblem for positions located close to grid cell borders,where there may be a considerable likelihood that thetrue position was located in the adjacent square.

Model. Based on the number of uplinks received bya single satellite during a single passage and on theinternal consistency of the solutions from the position-ing algorithms, each Argos position is assigned byService Argos to 1 of 6 precision classes: B, A, 0, 1, 2and 3 (Keating 1994). Location class B contains theleast precise positions and Location class 3 the mostprecise positions. Service Argos only specifies theprecision of the three best location classes (Locationclasses 1 to 3). Errors are given as 68% percentiles onlongitude and latitude. The 68% percentile for Loca-tion class 3 is 150 m, indicating that the reported lon-gitude is within 150 m from the true longitude for68% of the positions and likewise for latitude. Thisresult is not equivalent to 68% of the positions beingwithin 150 m of the true position, as errors on longi-tude and latitude are independent of one another. Thefraction of positions within 150 m of the true positionis significantly smaller, below 40%. The positioningerrors (expressed as 68% percentiles) for all 6 locationclasses were determined empirically by Vincent et al.(2002), see our Table 2. Note that Location class A isconsiderably more accurate than Location class 0 andcomparable to Location class 1.

We used the values measured by Vincent et al.(2002) below, including their results for Location class1 to 3, even though their estimates are not identical tothose supplied by Service Argos. Vincent et al. (2002)used transmitters with comparable specifications tothose used in our study and deployed in a similar

fashion (glued onto the fur of unrestrained grey sealsheld in captivity). We used the error estimates byVincent et al. (2002), but it would also be feasible touse other error estimates, if such exist and are consid-ered more appropriate for other applications.

Following Vincent et al. (2002), we filtered the datato remove extreme positions using a SAS-program(Argos_Filter V 7.01; prepared by Dave Douglas,USGS, Alaska Science Center). The program identifiesoutlying positions based on a maximum swimmingspeed between positions (McConnell et al. 1992) andangle between straight-line paths connecting 3 con-secutive positions (Keating 1994). The threshold valueswere a maximum swimming speed of 2.7 m s–1

(McConnell et al. 1992) and minimum angle of 10°.After filtering, it was assumed that the errors in lati-tude and longitude were normally distributed accord-ing to a bivariate Gaussian distribution. The bivariateGaussian distribution has the parameters μx and μy

(mean error on longitude and latitude, respectively), σx

and σy (standard deviation of errors) and ρ (correlationbetween errors on longitude and latitude). We furtherassumed that errors in longitude and latitude wereuncorrelated (ρ = 0) and that mean errors were zero (nosystematic bias in observed positions towards any par-ticular direction). These assumptions are all supportedby the results of Vincent et al. (2002).

The probability that the true position associated witha given observed position is located inside a given rec-tangle was assumed to depend on the distancebetween the observed position and the centre of therectangle as well as the size of the rectangle. The prob-ability that the true position of a seal is inside a partic-ular 10 × 10 km UTM square is calculated on the basisof the cumulated bivariate Gaussian distribution(Eq. A4; see also Fig. 2. Probabilities are shown as afunction of the distance between the centre of thesquare and the position reported by Service Argos(referred to hereafter as ‘observed position’). For thegood location classes (A, and 1 to 3) there is a very highlikelihood that the seal was in fact inside the square ofthe observed position, unless the observed positionwas within 1000 m of the border of the square. ForLocation classes 0 and B there is a significant probabil-ity that the true location was located in one of the adja-cent squares, even if the observed position was locatedin the centre of the square (probability of the true posi-tion being in an adjacent square is 0.10 and 0.38 forLocation class 0 and B, respectively).

The measure we are seeking is the combined proba-bility that at least 1 true position (n) was located insidea given square (A), P(nA > 0). P(nA > 0) is calculatedfrom the probability that each individual positionbelongs inside the square, both for observed positionsinside the square and in adjacent squares.

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Location Service Argos Vincent et al. (2002)class long. and lat. (m) Long. (m) Lat. (m)

3 150 295 1572 350 485 2591 1000 1021 4270 – 3029 1851A – 909 678B – 4815 3193

Table 2. 68% percentiles for difference between true and cal-culated position for each of the 6 location classes (for detailssee ‘Materials and methods; Model’). Nominal values pro-vided by System Argos, and empirically determined errorsfrom Vincent et al. (2002) are shown. –: no data available;

long.: longitude; lat.: latitude

Endang Species Res 4: 113–122, 2008

(1)

P(nA > 0) is calculated as 1 minus the probability thatno uplinks originated from the square (A). The proba-bility that no uplinks came from A is again calculatedas the product of the N probabilities that each of the Nindividually observed positions did not originate in theparticular square, 1–P(νi ∈ A). νi is the ith position inthe dataset, which contains a total of N positions, andnA is the number of true positions inside square A.

Consider as an example Square LH41 in Fig. 1. LH41contains 1 position of Location class 0 and 2 positions ofLocation class B. Fig. 3 shows the total probability thatat least 1 of these 3 positions originated from SquareLH41 (p = 0.922), as well as the probability for eachadjacent square that at least 1 of the positionsobserved inside the adjacent squares belonged inLH41. There is more than 50% chance that at least 1 ofthe positions in LH40 originated in LH41 and the sameis true for Square LH51. If all positions in the 9 squaresare included in the calculation, the total probabilitythat at least 1 position had its true origin inside LH41becomes: p(nLH41 > 0) = 0.985. In other words: the prob-ability that at least 1 seal visited LH41 and transmittedsignals while there is 98.5%.

A second parameter that can be associated with eachUTM-square is the most likely number of true positions

P n P n P AA A ii

N

( ) ( ) ( )> = − = = − − ∈[ ]=

∏0 1 0 1 11

ν

116

0 5 10 150

0.5

1

A

East/west displacement from centre (km)

Pro

bab

ility

A0123

Location class

B

B

North/south displacement from centre (km)

Pro

bab

ility

0 5 10 150

0.5

1

Diagonal displacement from centre (km)

Pro

bab

ility

0

0.5

1

0 5 10 15

C

Fig. 2. Calculated probability that the true position of thetransmitter associated with a particular position provided byService Argos belongs inside a 10 × 10 km Universal Trans-verse Mercator (UTM) square as a function of the distance be-tween the centre of the square and the observed position. (A)Displacement in longitude, (B) displacement in latitude and(C) displacement along the diagonal. Inside the grey rectan-gle the position reported by Service Argos is located in theUTM square being evaluated; in the white rectangle theobserved position is in the adjacent square. Small insertshows direction of displacement from centre of the grey

square. See Table 2 for location class

Easting (km)

330 340 350 3606200

6210

6220

6230

0.095

<0.001

–

0.562

0.922

–

0.006

0.504

<0.001

LH32

LH31

LH30

LH42

LH41

LH40

LH52

LH51

LH50

Nor

thin

g (k

m)

Fig. 3. Evaluation of Universal Transverse Mercator (UTM) 10× 10 km Square LH41 and adjacent squares. The probabilitythat at least 1 of the positions inside each square belongsin LH41 is calculated; see Appendix 1 for details on calcula-tion. No probabilities assigned to LH32 and LH42 as theycontained no observed positions. Individual positions are

indicated by black dots

Tougaard et al.: Argos telemetry and harbour seal abundance

inside the square, E(nA). This is given as the sum of theindividual probabilities associated with individualobservations:

(2)

Thus E(nA) is found for square A as the sum of the in-dividual probabilities that the ith position νi truly be-longs in the square, summed over all N positions of thedataset. E(nA) thus represents the expected averagenumber of positions in the square in the entirely hypo-thetical situation where all positions could be recalcu-lated based on the same number of uplinks from thesame true positions. In other words, E(nA) is the bestestimate of the number of true positions in the square.

Practical considerations. Service Argos suppliespositions in degrees longitude and latitude and theymust be converted to UTM coordinates by an appro-priate transformation, provided by GIS-software.P(nA > 0) and E(nA) can then be calculated square bysquare from Eqs. (A5) to (7). Positions may cover morethan 1 UTM zone, in which case calculations must beperformed separately for each of the zones covered,with a sufficiently large overlap. Difficulties arisewhere 2 UTM zones meet, as the squares are no longertrue squares but trapezoids and of unequal size. As Eq.(A5) can only handle rectangles, some approximationusing smaller rectangles must be made. In our study,most positions were located in UTM Zone 32N, withthe remaining positions in the eastern part of Zone31N. The overlap of positions into Zone 31N was con-sidered sufficiently small to justify a westward exten-sion of Zone 32 to contain all positions.

Eqs. (A5) & (7) include all positions in the dataset incalculation of P(nA > 0) and E(nA) for every grid cell.Calculations can be drastically reduced if more remotepositions are excluded. Fig. 2 shows that separation be-tween observed and true position very rarely exceeds15 km. This implies that for all practical purposes, thetrue position is either located in the same square as theobserved position or in one of the 8 immediately adja-cent squares, reducing computations considerably. Italso implies that if a square is empty and surrounded byonly empty squares then both P(nA > 0) and E(nA) canbe set to zero for all practical purposes.

The size of the grid cells was set at 10 × 10 km butboth larger and smaller cell sizes can be used. Smallercells, down to 4 × 4 km have been used with goodresults for smaller subsamples of the dataset. Decreas-ing cell size beyond this will not give increased resolu-tion, as resolution then becomes limited by the posi-tioning errors.

Positions on or close to land pose a separate problem.Positions on land should definitely be included in theanalysis if they are no more than 10 km from the shore.

In fact, observed positions in squares completely cov-ered by land should add more weight to the nearestsquare containing shoreline, as we know with cer-tainty that the true position was not in the square of theobserved position. Such a correction has not beenattempted here as it is not trivial and it was consideredthat it would add little to the quality of the results. Itmay however, significantly improve the quality of theresults in other locations, e.g. where animals are swim-ming in narrow waters between larger islands orpeninsulas.

If results are used in an atlas survey for which themethod was developed, a criterion for when to includecells as positive for presence is required. A criterionof 95% probability was used for the atlas of Danish mam-mals, i.e. all 10 × 10 km grid cells with a value of P(nA > 0)larger than 0.95 was counted as positive for presence.

RESULTS

Fig. 4A shows all positions of the dataset in the Ger-man Bight and central North Sea, together with high-lighting of 10 × 10 km UTM squares which containedone or more high-precision positions (Location class 1,2 or 3). This map clearly illustrates the large amount ofinformation lost if low precision positions are excludedfrom analysis.

Fig. 4B shows UTM squares coloured according tothe value of P(nA > 0). This map expresses the likeli-hood that one or more of the tagged seals visited theindividual squares, and thus reflects the area used bythe seals in the period the transmitters were active.

A large clustering of squares with very high proba-bilities of presence is found around the tagging siteand to the north and west of the site. Numerous posi-tions were received from this area, which containsimportant haulout sites and likely also foraging areas.High probability squares are also found further off-shore, reflecting the fact that several of the individualseals ventured out in a north-western direction to thecentral North Sea and spent considerable time at moreor less the same position, presumably foraging, beforethey found a different site or returned to the hauloutsites in the Wadden Sea. One seal swam to the Nether-lands and spent more than 1 mo around the island ofTerchelling, which is also reflected in high probabilitysquares in this area.

Values of E(nA) for UTM squares in the GermanBight and central North Sea are shown in Fig. 4C. Thismap represents the density of true positions and is thusa reflection of where the seals spent most of their time,although the probability of receiving locations may beinfluenced by differences in the surface behaviour ofthe seals among the different areas.

E n P AA ii

N

( ) ( )= ∈=

∑ ν1

117

Endang Species Res 4: 113–122, 2008118

9°6°E

56°N

54°

9°6°E

56°N

54°

9°6°E

56°N

54°

9°6°E

56°N

54°

LC 1, 2 and 3

n > 1

n = 1

P(nA > 0)

0.00 - 0.25

0.26 - 0.50

0.51 - 0.75

0.76 - 0.95

0.96 - 1.00

E (nA)

0.00 - 0.50

0.51 - 1.00

1.01 - 5.00

5.01 - 10.00

> 10.00

Kernel densities

25 %

50 %

75 %

95 %

A

C

B

D

Fig. 4. Satellite-derived positions of 10 harbour seals tracked in 2002 (indicated by black dots). (A) Universal Transverse Mercator(UTM) squares (10 × 10 km, Zone 32N) with high precision positions (LC, Location class 1, 2 or 3, see ‘Materials and methods’); (B)UTM squares with high probabilities of at least 1 true position (n) being inside a square (P(nA > 0)); (C) UTM squares with highexpected number of true positions (E(nA)); (D) Fixed kernel probabilities calculated from all positions calculated in ArcGIS animal

movement module, quartic kernels, scaling factor 106 and smoothing factor 3 × 105. See ‘Results’ for further explanation

Tougaard et al.: Argos telemetry and harbour seal abundance

Squares around the tagging site and other hauloutsites have very high values of E(nA), due to the largenumber of uplinks received from these areas. Thisresult reflects the fact that the seals clustered on a lim-ited number of sites during haulout, whereas they dis-tributed more widely during offshore foraging. Anadditional factor is that transmission conditions aremore favourable when the animals are on land or inshallow water (i.e. the transmitter is out of the watermore often). Nevertheless, there are several clusters ofpositions in the open sea where E(nA) is considerablyhigher than 1, indicating that these areas were usedextensively by 1 seal or more.

For comparison, contours of fixed kernels calculatedfrom the same dataset are shown in Fig. 4D. The 25%kernel density contour estimates the smallest area con-taining 25% of the positions and similarly for the 50, 75and 95% contours.

DISCUSSION

The method presented here for analysing Argospositional data was developed for a specific purpose,namely the collection of verified observations of sealsin connection with an atlas survey. For this purpose,the method worked very well. The method is simpleand no assumptions regarding the behaviour of thetagged animals are needed for the evaluation.

The method has potential value in other types of stud-ies, as in the context of evaluating effects of man-madeor natural structures on animal distribution, such aswhether animals pass under bridges, through narrowstraits or canals or spend time inside well defined areassuch as large harbours, offshore wind farms or marineprotected areas. In the absence of other data such as vi-sual sightings, Argos positions, even of poor precisionclasses, can thus help provide answers to the question:Do we have any evidence that species A has ever beenpresent in this particular area of interest? The methodmay even be applicable to other types of inaccurate po-sitional data where the precision of each position isknown, such as those obtained by acoustic positioningof sound sources (e.g. Wahlberg et al. 2001).

As stated in the ‘Introduction’, Argos based teleme-try systems are excellent for large-scale tracking ofmarine mammals, but at smaller scales the precision ofpositions is often too low to allow for strong conclusionson the exact location of the tagged animal. Problemsare exacerbated by the fact that most positionsreceived from transmitters on marine mammals arebased on very few uplinks and are consequently of lowprecision. Statistical methods are often needed inorder to interpret results at scales of 10 km and less.The method presented in this paper represents one

such approach, aimed at a very well-defined problem:How likely is it that tagged animals visited one or moreparticular areas? This goal was achieved by providingan objective method for evaluating Argos positioningdata in a 10 × 10 km UTM grid. There are fundamentallimitations of the Argos system, however, which arenot easy to overcome. One such limitation is that calcu-lations are based on uplinks, which are highly irregu-larly spaced in time as they are neither transmitted norreceived at regular intervals. Transmission rate isaffected by the behaviour of the animal and receptionis determined by time of satellite passages, as well asother factors such as waves and nearby high land thatcan obstruct the line of sight to satellites. Thus, themaps produced by a simple analysis of the positions,such as in the present study, must not be interpreteddirectly in terms of animal densities or probability ofpresence. Strictly speaking, the maps only expressuplink probabilities. That positions are concentratedaround a few haulout sites is not only due to the 10tagged seals converging on these sites, but also to thefact that the likelihood of receiving a position fromthese areas is very high, as haulout behaviour andswimming in shallow water favour transmission.

The most important limitation of the Argos positionsregards absence data, which in part results from theuneven distribution of positions in time. Thus, we can-not strictly know whether empty areas were visited bythe seals or not. The problem is evident in Fig. 4B,Cwhere empty squares are found among visitedsquares, as e.g. in the Dutch Wadden Sea and thewestern part of the study area. In cases where weknow that the seals must have passed through theseareas, we have no way of determining what route theytook. Addressing this would require that positions areevaluated as sequences and not individually. It is pos-sible to do this, but at the cost of introducing a numberof variables describing the statistics of animal move-ment (e.g. average and maximum swimming speed,directionality of movement). As the average swimmingbehaviour of the animals is likely to change, depend-ing on the activity of the animal (foraging, travellingor hauling out), a considerably more complex modelwould be the result, relying on many critical assump-tions and hence higher uncertainty in the results.

Comparison to kernel density estimates

The present method of evaluating ARGOS positionaldata has some resemblance to maps of kernel densityestimates (e.g. Worton 1989, Seaman & Powell 1996;see also our Fig. 4D) as they both evaluate the taggedanimals’ use of the area by assessing probabilities ofpresence. There are important differences, however.

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Endang Species Res 4: 113–122, 2008

First of all, a kernel density analysis assumes that allpositions are accurate, which is not correct. Ignoringthe inaccuracy of positions is likely to result in an over-estimation of home range sizes, especially whenanalysing at smaller scales. For example, consider thecase where an animal remains at the same spot for along time (e.g. during haulout). Received positions willbe distributed around this fixed true position and mayeasily show an inaccuracy of several km due to theuncertainty in the Argos positioning, resulting in aconsiderable kernel area, although the animal did notmove at all. Thus, caution should be exercised whenevaluating kernel density estimates on scales compa-rable to the size of the positioning errors.

Likewise, the present method cannot overcome theinaccuracy of the positions. There is also a limit tothe resolution with which data can be analysed in ameaningful way. As resolution is increased, by eva-luating smaller and smaller squares, the probabil-ities assigned to the individual squares also becomesmaller and smaller, and at a resolution roughly equiva-lent to the magnitude of the errors (i.e. a few km) nothingfurther is gained by decreasing the size of the squares,except a dramatic increase in computation time.

A more fundamental difference between kernel den-sity estimates and the present analysis is the smooth-ing involved in computation of kernel densities. Kerneldensity maps are created by the summation of a largenumber of smaller kernels each centred on 1 satelliteposition and all of the same size. The size of this funda-mental kernel is a determining factor for the outcomeof the final result, and changing this value, sometimestermed ‘smoothing factor’ will critically influence theresulting contours. A few scattered positions and ahigh smoothing factor will often result in contiguouskernel density contours that contain all positions, aswell as a great deal of the surroundings. This approachcan result in misleading conclusions, unless these lim-itations are kept in mind. As an example, consider afew positions evenly spaced more or less along astraight line, a pattern often seen for marine mammalsmoving from one area to another. A kernel density esti-mate based on these positions and a high smoothingfactor will tend to result in an oval shaped surface,which may extend many km to the sides of the track.Thus, although we have good reason to believe thatthe animal moved in a straight line, a kernel densityanalysis would suggest that there was a significantprobability that the animal could be found far to thesides of the track.

A low smoothing factor, on the other hand, will oftenresult in maps with many non-contiguous kernel densitycontours, each containing only a few or even only 1 posi-tion. Picking an appropriate amount of smoothing is thuscritical to the final result of the analysis.

No smoothing factor is included in the new grid-based method presented here, and areas with few orno positions will always result in low probability ofpresence. In some cases, such as in an atlas survey, thisis a considerable strength of the method, compared tokernel density maps, as no parameters other thanprecision of the satellite positions are needed forthe analysis. For other applications, some degree ofsmoothing may in fact be desirable, in which case ker-nel density maps should be favoured. Thus, it is notappropriate to generalise about whether kernel homeranges or the present grid analysis is superior. Resultsfrom both methods should be interpreted carefully andcautiously, especially on fine scales. For some applica-tions, they may supplement each other well. However,one case, in addition to the atlas survey, where the pre-sent method seems considerably more appropriatethan kernel density estimates is in spatial modelling ofanimal distribution.

Spatial modelling

Spatial modelling has become a central tool in thedescription of animal distribution in space (e.g. Elith etal. 2006, Redfern et al. 2006). In general terms, spatialmodelling is a set of statistical techniques used to cor-relate observations of animals in space with a range ofenvironmental predictors, whereby otherwise hiddenrelationships between predictors and animal occur-rence can be revealed. There are several problemsassociated with spatial modelling, particularly whenthe input is positional data from satellite telemetry.However, much of the groundbreaking work in thisfield has been on material from museum collections.Such data suffer from some of the same weaknesses astelemetry data, such as the low number of observa-tions, no information on absence and strong bias insampled areas. Addressing these issues has been acentral priority in the development of spatial modellingtools developed in recent years, and these models havebeen shown to perform well, even with strongly biaseddata (Elith et al. 2006).

Individual Argos positions are not particularly wellsuited as input for spatial modelling. Besides the prob-lem of inaccuracy of the positions, which one canchoose to ignore or solve by using the good locationclasses only, the main problem is how to include infor-mation on environmental conditions in areas where theanimals were not seen. This can be solved by the inclu-sion of random sampled points from low density areas;however, this approach is unsatisfactory, as it involvessubjective decisions on what areas to sample and howmany absence data should be included. The grid-basedapproach described here represents an alternative

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Tougaard et al.: Argos telemetry and harbour seal abundance

without the need for subjective decisions of that sort. Bydirectly evaluating presence in grid cells instead of us-ing the individual points, not only is the absence infor-mation included automatically, but the presence infor-mation is also supplied in a graded measure (asprobabilities, or expected number of true positions), in-stead of the binary presence/absence data from the po-sitions on their own. Thus, the grid analysis methodprovides output of a nature which can be directly en-tered into spatial modelling algorithms and, thus, haspotential for becoming a valuable tool to determinespatial distribution from satellite-derived positions.

Acknowledgements. Capture and tagging of seals was donewith the help of the Search and Rescue boat of Rømø Islandand the project would not have been possible without theskilled assistance of the crew. Staff from the Fisheries andMaritime Museum and the National Environmental Re-search Institute, as well as a large number of volunteers, areacknowledged for enthusiastic participation in the field. MaryS. Wisz is thanked for fruitful discussions and suggestionsregarding this manuscript. Rune Dietz, Susi Edrén and KasperJohansen are all thanked for assisting in analysis of the dataand for preparing final versions of the maps. Suggestions from3 anonymous reviewers were helpful in improving the manu-script. The satellite telemetry study was funded by the DanishEnergy Authority through contract with Elsam A/S as part ofa monitoring program on seals in and around the Horns RevOffshore Wind Farm.

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Vincent C, McConnell BJ, Ridoux V, Fedak MA (2002) Asses-ment of Argos location accuracy from sattelite tagsdeployed on captive gray seals. Mar Mamm Sci 18:156–166

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Endang Species Res 4: 113–122, 2008122

Given is an ARGOS position νobs with coordinates (xobs, yobs) and a corresponding true position νtrue. The 2 positions are con-nected through the positioning errors εx and εy:

(A1)

The probability that a given point (a,b) is the true position corresponding to νobs is given as:

(A2)

The probability in Eq. (A2) is described by the bivariate Gaussian distribution if εx and εy are normally distributed:

where

and

μx and μy are mean errors, σx and σy are standard deviations and ρ the correlation between εx and εy. It is reasonable to assumethat errors on longitude and latitude are uncorrelated (ρ = 0) and that mean error is zero (observed positions are not displacedfrom true positions in a systematic direction). These assumptions are supported by results of Vincent et al. (2002). Given theseassumption, the above reduces to:

(A3)

In order to calculate the probability that the true position νtrue corresponding to the observed position νobs is located insidesome square A (or rectangle), we introduce the standardized cumulated probability function for the uncorrelated bivariateGaussian distribution Φ (ρ = 0; μx = μy = 0; σx = σy = 1):

(A4)

As for the univariate distribution, this integral cannot be solved analytically, and one is forced to resort to polynomial approx-imation or numerical methods. The approximation given by Divgi (1979) was used in this example. The probability we arelooking for is the probability that the true position νtrue corresponding to the observed position νobs with coordinates (xobs,yobs)is located inside a rectangle A with corners (a1,b1) and (a2,b2). This is given as:

(A5)

Eq. (A5) gives the probability that a single position belongs in the area A. By combining the results for all positions receivedwe can calculate the probability that the number of true positions inside a given grid cell is larger than zero (P(nA > 0)). Thisprobability is easily calculated from the joint probability that none of the N positions in the dataset belongs to A (P(nA = 0)):

(A6)

A second useful parameter is the most likely number of true positions in A. This figure equals the mean number of true posi-tions in A in the hypothetical situation where the entire data collection is repeated several times. E(nA) is given as:

(A7)E n P AA ii

N

( ) ( )= ∈=

∑ ν1

P n P n P AA A ii

N

( ) ( ) ( ( ))> = − = = − − ∈=

∏0 1 0 1 11

ν

P Ax y a

ax x

( )trueνπσ σ

σ∈ = ∫

− −⎛⎝⎜

⎞⎠⎟ +

12

1

2

212

e

obs

x

yy y

b

b

x y

a x

−⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∫ =

−

obs

y

d d

obs

σ

σ

2

1

2

1Φxx y x y

b y a x b y, ,1 2 2−⎛

⎝⎜⎞⎠⎟

+ − −⎛⎝⎜

⎞obs obs obs

σ σ σΦ

⎠⎠⎟− − −⎛

⎝⎜⎞⎠⎟

+ −Φ Φa x b y a x b

x y x

1 2 2obs obs obs

σ σ σ, , 11 −⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

y

y

obs

σ

Φ( , ) ( )a b P x a y b x yx yba

= < ∧ < =− +

−∞−∞∫∫

12

2 2

2

πe d d

P x yx y

x

x

y

y( , )ε επσ σ

εσ

εσ=

− +⎡⎣⎢

⎤⎦⎥1

2

2

2

2

2e

12

ρσ

σ σ= xy

x yz x x

x

y y

y

x x y y= − +−

−− −( ) ( ) ( )( )ε μ

σε μ

σρ ε μ ε μ

σ

2

2

2

2

2

xx yσ

P x yx y

z

( , ) ( )ε επσ σ ρ

ρ=−

−−1

2 1 22 1 2

e

P x a y b P x a y b

P

x y( ) ( )true true obs obs= ∧ = = − = ∧ − = =ε ε(( )ε εx yx a y b= − ∧ = −obs obs

εε

x

y

x x

y y

= −= −

obs true

obs true

Appendix 1. Calculation of P(nA > 0) and E(nA)

Editorial responsibility: Helene Marsh,Townsville, Queensland, Australia

Submitted: March 12, 2007; Accepted: November 5, 2007Proofs received from author(s): December 21, 2007