A Voyage to Africa by Mr Swift

Niklas Wahlström, Fredrik Gustafsson and Susanne Åkesson

Linköping University Post Print

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Niklas Wahlström, Fredrik Gustafsson and Susanne Åkesson, A Voyage to Africa by Mr

Swift, 2012, Proceedings of the 15th International Conference on Information Fusion

(FUSION), 808-815.

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-85905

A Voyage to Africa by Mr SwiftNiklas Wahlstrom and Fredrik Gustafsson

Division of Automatic ControlLinkoping University

Email: {nikwa, fredrik}@isy.liu.se

Susanne AkessonDepartment of Biology

Centre for Animal Movement ResearchLund University

Email: susanne.akesson@biol.lu.se

Abstract—A male common swift Apus apus was equipped witha light logger on August 5, 2010, and again captured in his nest298 days later. The data stored in the light logger enables analysisof the fascinating travel it made in this time period.

The state of the art algorithm for geolocation based on lightloggers consists in computing first sunrise and sunset from thelogged data, which are then converted to midday (gives longitude)and day length (gives latitude). This approach has singularitiesat the spring and fall equinoxes, and gives a bias for fast daytransitions in the east-west direction.

We derive a flexible particle filter solution, where sunset andsunrise are processed in separately measurement updates, andwhere the motion model has two modes, one for migration andone for stationary long time visits, which are designed to fitthe flying pattern of the swift. This approach circumvents theaforementioned problems with singularity and bias, and providesrealistic confidence bounds on the geolocation as well as anestimate of the migration mode.

Index Terms—nonlinear filtering, particle filter, geolocation,light levels

I. INTRODUCTION

Geolocation for learning migration behavior of animals is animportant area for ecologists and epidemiologists. With an everlasting improvement in sensor technology and miniaturizationof electronics, more and smaller species can be studied.More specifically, researchers are no longer limited to satellitebased geolocation, which requires substantial battery capacityand in practice also radio transmitters to communicate thegeolocation data, which in turn requires even more battery withthe consequence that only larger birds can be studied. One suchenabling technology is light loggers permitting documentationof migration by individual small songbirds, waders and swifts,which so far is not extensively studied. Commercial versionsinclude a light sensor, a battery, a memory and a clockenclosed in a small housing that can be fit to the tarsus ormounted with a harness to the back of the bird. As a rule ofthumb, the sensor can weigh at most 5% of a bird’s weight.State of the art sensors weigh 2 grams, and thus as smallbirds as a swift (40g) can be marked. The sensor unit cannotcommunicate, so this approach hinges on that the birds arecaught or found when they die, so the memory can be readoff.

The swift is an interesting bird, since it is believed to spendall of its life on the wings, except for the nesting period. Thismakes the light data particular good, since there is barelyany surrounding vegetation that disturb the measurements.Also the migration pattern of the swift is fascinating, since it

Fig. 1: A male common swift after his voyage to Africa andback. During the journey a light logger mounted to his backrecording the light intensity.

travels long distances over the seasons, but always returns tobasically the same place for mating in the summer. This uniquenavigation ability is believed to be genetically inherited, andit might have persisted since the last ice age. The swift inFigure 1 is a young male and one of the first marked swiftthat has been found. The researchers know very little about themigration pattern for swifts during the winter time in Africa,since there are almost no reports from Africa on found specieswith the classical ring.

The light logger samples light intensity every 5-10 minutes.It is quite a coarse information suffering from saturation atboth ends. Thus, it is only the transitions between nightand day and vice versa that contain information with thecurrent sensor. The theory of geolocation by light levels isdescribed in [1] for elephant seals, where sensitivity andgeometrical relations are discussed in detail. The accuracy ofthe geolocation is evaluated on different sharks by comparingto a satellite navigation system, and the result is shown to bein the order of 1 degree in longitude and latitude.

If the animal is known to be at rest during the night, thetwo positions corresponding to sunset and sunrise can beassumed the same, and the unknown longitude and latitudecan be solved from the two measurements uniquely. This isthe basis for the geolocation software that comes with thesensor. There is an obvious singularity for the two days of

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equinox when the sun is in the same plane as the equator, andthus the two manifolds in Figure 2 are vertical lines. In praxis15 days on each side of spring and autumn equinoxes areomitted from analyses [2]. Another problem occurs if the birdis moving between sunrise and sunset, in which case middayand daylength is shifted slightly causing a bias in the position.This is especially a problem during fast migration flights, whena swift may cover up to 650 km/day [3].

Fig. 2: Binary day-light model for a particular time t. Theshape of the dark area depends on the time of the year, andthe horizontal position of the dark area depends on the timeof the day.

We propose a nonlinear filtering framework, where thesunset and sunrise are treated as separate measurement up-dates, and an irregularly sampled motion model is used forthe time update. This removes the bias problem and also anoise correlation artifact. It also mitigates the singularity atthe equinox, where still useful positions can be computed. Themotion model has two modes, corresponding to stationary andmigrating flight. The filter thus has adaptive sensitivity, givinghigher position accuracy at the stationary mode.

Animal geolocation based on light levels can be tracedback to at least l986 [4]. Other publications such as [5], [1],[6] and [2] have also studied this problem. However, to thebest of the authors knowledge, this is the first time that thisapplications has been put into a statistical filtering frameworkusing separate measurement updates for sunset and sunrise.

The paper outline is as follows: In Section II an appropriatesensor and motion model for this application will be presentedand the state estimation algorithm is given in Section III.The paper is concluded with the results on real world datain Section IV followed by the conclusions in Section V.

II. MODELS

The mathematical framework can be summarized in a statespace model with state xk, position dependent measurementyk, process noise wk, and measurement noise ek:

xk+1 = f(xk) + wk, (1a)yk = h(xk) + ek. (1b)

The state includes position (Xk, Yk) encoded as longitudeXk and latitude Yk, a velocity (Xk, Yk) as well as a modeparameter δk.

A. Sunrise and Sunset Models

Figure 2 shows how the sunset and sunrise, respectively, ateach time defines a manifold on earth [7]. A sensor consistingof a light-logger and clock can detect these two events. Thebright part of the earth is limited by a great circle orthogonalto the sun at each time as depicted in Figure 4.

The time of the sunrise and sunset can easily be derivedwhen knowing the daylength and the time of the midday. Themidday will only depend on the longitude of the observer. Atlongitude Xk = 0◦ the midday occurs at 12.00 noon. Whengoing X = 360◦/24 = 15◦ east, the midday occurs one hourearlier at 11.00 a.m (earlier, since the sun rises in the east).This gives the relation

hmidday(Xk) = 12− 1

15Xk. (2)

Further, the daylength will depend on the latitude of theobserver as well as on the time of the year. The relation canbe derived by making use of the coordinate transformationfrom the equatorial coordinate system to the horizontal coor-dinate system. These coordinate systems are used for mappingpositions on the celestial sphere and can thus be utilized fordescribing the position of the sun.

The horizontal coordinate system uses the observer’s localhorizon as the fundamental plane and the position of the sun isdescribed with its altitude h above the horizon and its azimuthA measured from the south increasing towards the east, seeFigure 3.

South

Altitude h

Azimuth ASun

Horizon

Fig. 3: The horizontal coordinate system.

In the equatorial coordinate system the fundamental planeis defined by the Earth’s equator. Here, the position of the sunis described with the solar hour angle H expressed in angularmeasurement from the solar noon, and the declination of thesun d.

All of these four angles are defined on the celestial sphere[7], [8]. However, in this work we are interested in describingthe position of the observer on the earth rather than the positionof the sun on the celestial sphere. Therefore, in Figure 4,the corresponding angles on the earth are depicted. Here,the altitude h has the interpretation of being the orthogonaldistance (measured in degrees) to the great circle separatingthe bright and dark part of the earth, the hour angle H isthe longitude relative to the solar noon meridian and thedeclination d is the tilt of the earth’s axis towards the sunas well as the latitude where the sun reaches its zenith.

Following [7], these two coordinate systems are related as

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Fig. 4: The hour angle H , declination d and altitude hprojected on to the earth, together with the observer’s latitudeY . The position ”zenith” is the location where the sun is at itszenith. The geometry of these angles gives the relation (3a).

sinh = sinY sin d+ cosY cos d cosH (3a)

tanA =sinH

cosH sinY − tan d cosY, (3b)

where Y is the latitude of the observer. The declination of thesun can also be seen as the angle between the Earth’s axis anda line perpendicular to the Earth’s orbit. Thus, the declinationwill change over the year and is given by

d(tk) = −23.439◦ cos

(360◦

365.25tk

), (4)

where tk is the number of days after the winter solstice. Sincea day of 24 hours corresponds to an hour angle of H = 360◦,(3a) gives us the relation

hdaylength(tk, Yk) =24

360◦arccos

(sinh0 − sinYk sin d(tk)

cosYk cos d(tk)

).

(5)

The altitude h0 is here considered as a constant. It representsthe geometrical altitude of the sun at the time apparent risingand setting. Due to atmospheric refraction, these events occuralready when the sun geometrically is below the horizon at analtitude of h0 = −0◦.83 [7]. Furthermore, the light intensitystarts increasing before the sunrise and is still increasingafter the event. Noticeable is that the most distinct transitionbetween day and night occurs when the sun is about 6◦ belowthe horizon [1]. Thus, the altitude h0 will in practice dependon how the light intensity data is thresholded.

Note that if the argument of arccosine in (5) is larger than1 in absolute value, the sun will remain either above or belowthe horizon the whole day.

With this information we can define a measurement modelfor sunrise and sunset respectively

yrise(tk) = hrise(tk, Xk, Yk) + erisek , (6a)yset(tl) = hset(tl, Xl, Yl) + esetl . (6b)

where

hrise(tk, Xk, Yk) = hmidday(Xk)− hdaylength(tk, Yk)

2(6c)

hset(tl, Xl, Yl) = hmidday(Xl) +hdaylength(tl, Yl)

2(6d)

B. Sensor Error ModelThe errors erise(tk) and eset(tl) for sunrise and sunset,

respectively, consist of different kinds of errors:1) Detection errors from the light logger data. The data

in Figure 8 indicates that this error is Gaussian with astandard deviation of slightly more than four minutes.Note that four minutes corresponds to 1 degree, whichis 120 km in north-south direction and 120 times cosineof latitude in east-west direction.

2) Position dependent variations. For instance, the days arelonger over sea than land [1]. We will neglect this error,since the bird appears to be over land most of the time.It would be no problem to cover this in our framework,though.

3) Also the latitude and time of year may affect the error.This is a subject for future studies.

4) Weather dependent variations, where sunny days arelonger than cloudy ones. This can be incorporated byusing data from historic weather data bases in ourframework, but this is also a subject for future studies.

Note that the errors in sunset and sunrise can be seen asindependent, and thus the error in day length and midday areactually correlated.

C. Kinematic ModelThe kinematic model of migrating birds is characterized by

two modes consisting of a stationary mode on their breeding,wintering or moulting sites, as well as a migration mode [9]. Inorder to describe the transition from the stationary mode to themigration mode a transition mode is used. The mode parameterδk ∈ {stationary mode, transition mode, migration mode} ishere modeled as a hidden Markov state with a specifiedtransition probability Πk resulting in

p(δk+1|δk) = Π(δk+1,δk)k . (7)

In Figure 5 the corresponding Markov chain is drawn.In the stationary mode it is sufficient to model the dynamics

of the swift with a constant position model, which uses a twodimensional position

x(t) =

(X(t)Y (t)

), x(t) =

(wX(t)wY (t)

)(8a)

The corresponding discrete time model is given by

xk+1 =

(1 00 1

)xk +

(T 00 T

)(wXkwYk

). (8b)

810

where the input wk is considered as random process noise. Inthe migration mode the velocity of the bird will be of greatimportance when predicting the next position. This can becaptured by using a constant velocity model. This is still afairly simple motion model, yet one of the most common onesin target tracking applications where no inertial measurementsare available.

x(t) =

X(t)Y (t)

X(t)

Y (t)

, x(t) =

X(t)

Y (t)wX(t)wY (t)

(9a)

The corresponding discrete time model is given by

xk+1 =

1 0 T 00 1 0 T0 0 1 00 0 0 1

︸ ︷︷ ︸

=FCV

xk +

T 2/2 0

0 T 2/2T 00 T

︸ ︷︷ ︸

=GCV

(wXkwYk

).

(9b)

In a similar manner the constant position model (8b) can beredefined to include the velocity, ever though it is not used inthe position update

xk+1 =

1 0 0 00 1 0 00 0 1 00 0 0 1

︸ ︷︷ ︸

=FCP

xk +

T 00 T1 00 1

︸ ︷︷ ︸

=GCP

(wXkwYk

). (10)

In order to initialize the velocity correctly during the transi-tion from stationary to migration mode, a third transition modeis used to enlarge the covariance of the position. This modealso uses a constant position model but with a much largerprocess noise as for the stationary mode.

By modeling the process noise wk ∼ N (0, Qδk) to be whiteGaussian for the all modes, the transition density for xk andδk can be described as

p(xk+1, δk+1|xk, δk) = Π(δk+1,δk)k p(xk+1|xk, δk) (11)

where

p(xk+1|xk, δk) =N (FCPxk, GCPQ

sGTCP ), δk = stationary modeN (FCPxk, GCPQ

tGTCP ), δk = transition modeN (FCV xk, GCVQ

mGTCV ), δk = migration mode

where Qs, Qt and Qm is the process noise covariance for thethree modes. Note that we will will choose Qt � Qs in orderto capture the character of the transition mentioned above.

III. STATE ESTIMATION

The nonlinear filtering problem (1) will here be solved usinga marginalized particle filter. For this application (includingfuture extensions) this is a sound approach due to manyreasons:

δ = s

δ = t

δ = m

pst

1− pst

1

1− pmspms

Fig. 5: The Markov chain for the mode parameter δ. The modeδ = s is a the stationary mode, where xk constant positionmodel is used, δ = m corresponds to the migration mode usinga constant velocity model, δ = t corresponds to a transitionstate from δ = s to δ = m where a faster constant positionmodel is used.

• Like any filter, it can handle partial information of theposition, so it can process sunrises and sunsets separately.

• Also like any filter, it can handle multiple modes byrunning two or more filters in parallel, and fusing theirstates according to their performance.

• In contrast to other filters, it can handle multiple modesby augmenting the state vector with the mode parameter.This approach will be used in this work.

• The particle filter can handle multi-modal position dis-tributions better than any other filter, which is useful forrobust filtering where a lot of outliers in data occur (falseand missed detections from the logged light data).

• It can handle position dependent noise, like ground veg-etation type and local weather dependent noise distribu-tions.

• It can easily include state constraints, such as maximumspeed.

The time update of the position in the particle filter consistshere of the following steps:

1) Simulate N noise vectors wik ∼ N (0, Qδk).2) Propagate the set of particles according to (10) or (9b)

depending on δk.We will here make use of the fact that the sensor model (6)

depends on the position and the time only,

yk = h(tk, Xk, Yk) + ek. (12)

Since the motion models (10) and (9b) are linear in thestate and noise, the marginalized particle filter applies, sothe velocity component can be handled in a numerically veryefficient way.

Let pk = (Xk, Yk)T and vk = (Xk, Yk)T . Then, constantvelocity model (9b) and the sensormodel (12) can be rewrittenas

pk+1 = pk + Tvk +T 2

2wk, (13a)

yk = h(tk, pk) + ek, (13b)vk+1 = vk + Twk, (13c)

pk+1 − pk = Tvk +T 2

2wk. (13d)

811

We here use the particle filter for (13ab) and the Kalmanfilter for (13cd). Note that (13ad) are the same two equations,interpreted in two different ways. The time update in theparticle filter becomes

wik ∼ N(0, Qk

), (14a)

pik+1 = pik + Tvik +T 2

2wik. (14b)

Conversely, we use the position as a measurement in theKalman filter. For this particular structure, the general resultgiven in Theorem 2.1 in [10] simplifies a lot, and we get acombined update

vik+1|k =pik+1 − pik

T(14c)

Pk+1|k = Pk|k−1 − Pk|k−1(Pk|k−1 +

T 2

4Qk)−1

Pk|k−1.

(14d)

Note that each particle has an individual velocity estimatevik|k−1 but a common covariance Pk|k−1. Further, for a time-invariant Qk = Q, the covariance matrix converges quitequickly to Pk|k−1 = 0, and the Kalman filter is in fact notneeded and can be replaced with a deterministic update of thevelocity.

In a similar manner we get the same velocity update(14c) when using the constant position model (9b). Note thatthis velocity update is not needed affect the position if theparticle is in δk =”stationary mode”. However, in the modeδk =”transition mode” it is crucial since that velocity willbe used in the time update in the next time instance whenbeing in δk =”migration mode”. The estimation framework ispresented in Algorithm 1.

IV. RESULTS FROM REAL WORLD DATA

The proposed tracking framework has been validated onreal world data. This section presents this data as well as thetracking results.

A. The data

A light logger was mounted on a swift which was releasedfrom the very south of Sweden. Ten months later it wascaptured again at its own nest and the light logger was removedfrom the bird. The recorded data consists of light intensitymeasurements during a period of 298 days from 5th of August2010 to 29th of May 2011. From this data the universal timeof sunrise and sunset has been extracted by thresholding thedata, see [11] for further details on how this has been done.Thus, the preprocessed data consists of the universal time of298 sunrises and 298 sunsets as depicted in Figure 6.

From this information the daylength and midday can easilybe extracted as presented in Figure 7. The state of the artalgorithm is to use this information to convert to longitudeand latitude using (2) and (5). However, with Algorithm 1 wepropose to use time of sunrise and sunset to prevent the biasthat would occur in fast day transitions.

Furthermore, the daylength at the equinoxes gives us im-portant information about the data. An equinox occurs when

Algorithm 1 Migrating bird tracking using particle filterChoose number of particles N .Initialization: Generate pi1 ∼ pp0 , vi1 ∼ pv0 and δi1 ∼ pδ0 ,i = 1, · · · , N , encode the augmented state as xi1 = [xi1, δ

i1] =

[pi1, vi1, δ

i1], let ω0 = 1/N and k = 1.

Iteration over the days:For t = 1, 2, · · ·

Iteration over sunrise and sunset:For m = {sunrise, sunset}

1) Measurement update:For i = 1 : N

ωik =ωik−1p

m(yk|xik, tk)∑Nj=1 ω

jk−1p

m(yk|xik, tk)(15)

where yk = ym(t) is the time of sunrise/sunset atday t, measured in hours, and tk = t+yk/24 is thetime, measured in days.

2) Estimation: The state and covariance is estimatedby

xk =N∑i=1

ωikxik (16)

Pk =N∑i=1

ωik(xik − xk)(xik − xk)T (17)

3) Resample: Take N samples with replacement fromthe set {xik}Ni=1 where the probability to take samplei is ωik and let ωik = 1/N .

4) Time update:Compute the sampling interval:Tk = tk − tk−1

For i = 1 : N

For each part of the state vector xik = [pik, vik, δ

ik]

do the following:a) Generate predictions of the position depend-

ing on the modewik ∼ N (0, Qδ

ik)

If δik =”stationary mode” or δik =”transitionmode”pik+1 = pik + Tkw

ik

elseif δik =”migration mode”

pik+1 = pik + Tkvik +

T 2k

2 wik

b) Generate predictions of the modeδik+1 ∼ p(δk+1|δik)

c) Generate predictions of the velocityvik+1 =

pik+1−pik

Tk

5) Set k := k + 1

the declination of the sun (4) is equal to zero. Using (5), this

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2

4

6

8

10

12

14

16

18

20

08/05 09/23 Sep. Equinox

12/21 Dec. Solstice

03/20 Mar. Equinox

05/29

Tim

e [

h]

The data

Sunrise

Sunset

Fig. 6: Universal time of the sunrise and the sunset during the10 months journey of the swift

gives the simplified relation

hdaylength(Yk) =24

360◦arccos

(sinh0cosYk

)(18)

Thus, by assuming an altitude of h0 = 0 for sunrise andsunset, the daylength would be 12 hours all over the world.Further, since cosYk > 0, the sign of h0 will decide whetherthe daylength is longer or shorter than 12 hours. However,by consulting the data in Figure 7 it can be noticed that thedaylength is slightly longer than 12 hours at the Septemberequinox and slightly shorter than 12 hours at the Marchequinox, which would require a positive h0 and a negative h0respectively. As a compromise, we have here chosen h0 = 0.

10

11

12

13

14

15

16

17

08/05 09/23 Sep. Equinox

12/21 Dec. Solstice

03/20 Mar. Equinox

05/29

Tim

e [

h]

Midday and daylength

Daylength

Midday

Fig. 7: The daylength and the time of the midday during the10 months journey of the swift.

−20 −10 0 10 200

5

10

15

20

25

30

35Sunrise std=5.1127 min

Time [Min]

(a) Sunrise

−20 −10 0 10 200

5

10

15

20

25

30

35Sunset std=4.2798 min

Time [Min]

(b) Sunset

Fig. 8: Histograms for the detrended sunrise and sunsetmeasurements from the 10th of December to the 25th of Apriltogether with a Gaussian approximation.

From the data in Figure 6 the variance of the measurementnoise can be estimated. During the period from the 10th ofDecember to the 25th of April the data is fairly constant. (Aswe later will see in Figure 9 this corresponds to when the swiftis at its wintering site in Africa.) This data has been detrendedand is visualized as a histograms in Figure 8 for sunrise andsunset, respectively. From this data the standard deviation ofthe measurement noise can be estimated to approximately 5minutes for both sunrise and sunset.

B. Results

Algorithm 1 using N = 500 particles has been implementedand evaluated on the data presented in Figure 6. For the modeparameter δk the Markov chain in Figure 5 has been used withthe transition probabilities

pst = p(δk+1 = stat. m.|δk = trans. m.) = 0.2 (19a)pms = p(δk+1 = mig. m.|δk = stat. m.) = 0.05 (19b)

and

Qδk=stat. mode = 12 · I2 (20a)

Qδk=trans. mode = 102 · I2 (20b)

Qδk=mig. mode = 22 · I2 (20c)

Furthermore, according to Figure 8, the measurement noisehas a standard deviation of approximately 5 minutes. However,in order to compensate for the unknown bias caused by theunknown altitude h0 as explained in Section IV-A, a standarddeviation of 10 minutes has been used in the filter

eset ∼ N (0, (10/60)2), erise ∼ N (0, (10/60)2). (21)

The tracking performance is presented in Figure 9. Thetracking result will also be compared with a non-filteringsolution where we assume that the bird has not moved duringthe period from sunset to sunrise. Then, by using (6) andassuming (Xk, Yk) = (Xl, Yl) the longitude and latitude canbe solved uniquely each day separately. By using inversemapping we also get a corresponding covariance

xk = h−1(yk)

Cov(xk) =(∇hT (xk)R−1∇h(xk)

)−1

813

08/05

08/1909/0309/17

10/02

10/1610/31

11/1411/29

12/1312/2801/1101/2602/0902/2403/1003/25

04/0804/23

05/07

05/22

05/29

The tracked trajectory of the swift

15° W 0

° 15

° E 30

° E

15° S

0°

15° N

30° N

45° N

60° N

Fig. 9: The trajectory of the swift during a period of 298 days.The positions are estimated at each sunrise and sunset.

where

h =

(hsunrise

hsunset

), yk =

(ysunrise(tk)ysunset(tk)

)and xk =

(Xk

Yk

).

In Figure 10 the estimated position for the two methods ispresented with a 90% confidence interval together with theestimated mode parameter from the particle filter implemen-tation.

In Figure 11 two periods are zoomed in order to point outthe differences between the two methods. According to Fig-ure 11a, the particle filter implementation manages to mitigatethe singularity due to the September equinox. The variance isstill increasing for the particle filter implementation, howevernot as much as for the inverse mapping method. This is mainlydue to the fact that we use a motion model.

As explained earlier, fast transitions in east-west directionwill give rise to shorter/longer measured daylength. This willlead to a bias in the latitude estimation using the inversemapping method since it wrongly assumes that the light loggermeasures the sunrise and sunset at the same position. In Fig-

−50

0

50

Latitude

De

gre

e [

°]

08/05 09/23 Sep. Equinox

12/21 Dec. Solstice

03/20 Mar. Equinox

05/29

−20

−10

0

10

20

30Longitude

08/05 09/23 Sep. Equinox

12/21 Dec. Solstice

03/20 Mar. Equinox

05/29

De

gre

e [

°]

0

0.2

0.4

0.6

0.8

1White: Stationary mode, Black: Trasition mode, Grey: Migration mode

08/05 09/23 Sep. Equinox

12/21 Dec. Solstice

03/20 Mar. Equinox

05/29

Particle filter implementation

Inverse mapping

Fig. 10: The estimated position and mode of the swift duringa period of 298 days. The presented particle filter imple-mentation is here compared with using inverse mapping. Theestimates are presented with a 90% confidence interval.

ure 11b such a bias for the latitude can be seen during a periodaround May where the swift is making a fast transition fromAfrica back to Europe. The movement in west direction willmake the measured daylength longer than the actual daylengthat the corresponding latitude. For the inverse mapping methodthis will give a bias towards north since the daylength is longeron the northern hemisphere during the summer.

Finally in Figure 12 the trajectories for the particle filterimplementation is presented together with covariance ellipsesrepresenting the estimation uncertainty. Here also the positionof the start and end point of the journey is depicted.

V. CONCLUSION AND FUTURE WORK

In this paper, a particle filter solution has been presentedestimating the trajectory of a migrating bird using light loggerdata. A sensor model has been presented based on astronom-ical formulas consisting of a measurement update at sunrise

814

−50

0

50

Latitude

09/03 09/13 09/23 10/03

−20

−15

−10

−5

0

5

10Longitude

09/03 09/13 09/23 10/03

(a) September equinox

−10

0

10

20

30

40

50Latitude

05/11 05/16 05/21 05/26

0

5

10

15

20

25Longitude

05/11 05/16 05/21 05/26

(b) In the middle of May.

Fig. 11: The estimated position of the swift during a montharound the fall equinox when the swift is migrating fromEurope to the east of Africa as well as two weeks in themiddle of May. The presented particle filter implementation(blue) is here compared with using inverse mapping (green).The estimates are presented with a 90% confidence interval.

and sunset respectively, and a suitable motion describing theflying pattern of migrating birds. The implementation hasbeen validated on real data and compared with the state ofthe art algorithm based on a purely deterministic approach.The proposed solution outperforms the existing method inreliability during the equinoxes and removes problem with biasdue to fast day transitions. In addition, the proposed methodprovides an estimate of the migration mode suitable for furtheranalysis of the flying pattern of the bird. In the future, theframework will be extended to process the light intensity datadirectly as well as including position and weather dependentnoise and bias.

ACKNOWLEDGMENT

The authors gratefully acknowledge fundings from theSwedish Foundation for Strategic Research under the projectCooperative Localization.

REFERENCES

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[2] B. Stutchbury, S. Tarof, T. Done, E. Gow, P. Kramer, J. Tautin, J. Fox,and V. Afanasyev, “Tracking long-distance songbird migration by usinggeolocators,” Science, vol. 323, no. 5916, p. 896, 2009.

[3] S. Akesson, R. Klaassen, J. Holmgren, J. W. Fox, and A. Hedenstrom,“Migration route and strategies in a highly aerial migrant, the commonswift Apus apus, revealed by light-level geolocators.” Submitted.

[4] P. Smith and D. Goodman, “Determining fish movements from anarchival tag: Precision of geographical positions made from a timeseries of swimming temperature and depth,” US National Oceanic andAtmospheric Administration, Tech. Rep., 1986.

08/05

08/1909/0309/17

10/02

10/1610/31

11/1411/2912/1312/2801/1101/2602/0902/2403/1003/25

04/0804/23

05/07

05/22

05/29

A Voyage to Africa by Mr Swift

15° W 0

° 15

° E 30

° E

15° S

0°

15° N

30° N

45° N

60° N

Sunrise

Sunset

Start and end position

Fig. 12: The trajectory of the swift during a period of 298days. The positions are estimated at each sunrise and sunset. Atevery 5th day, the accuracy of position estimate is visualizedtogether with a 90% confidence interval.

[5] R. Wilson, J. Ducamp, W. Rees, B. Culik, and K. Niekamp, “Estimationof location: global coverage using light intensity,” Wildlife telemetry:Remote monitoring and tracking of animals, pp. 131–134, 1992.

[6] P. Ekstrom, “An advance in geolocation by light,” Memoirs of theNational Institute of Polar Research, vol. 58, pp. 210–226, 2004.

[7] J. Meeus, Astronomical algorithms. Willmann-Bell, Incorporated, 1991.[8] O. Montenbruck and T. Pfleger, Astronomy on the personal computer.

Springer Verlag, 1991, vol. 1.[9] I. Newton, The migration ecology of birds. Academic Pr, 2008.

[10] T. Schon, F. Gustafsson, and P.-J. Nordlund, “Marginalized particlefilters for mixed linear/nonlinear state-space models,” IEEE Transactionson Signal Processing, vol. 53, no. 7, pp. 2279–2289, Jul. 2005.

[11] J. Fox, “Geolocator manual,” 2010.

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