the problem of estimating wind drift in migrating birds

J. theor. Biol. (2002) 218, 485–496doi:10.1006/yjtbi.3094, available online at http://www.idealibrary.com on

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The Problem of Estimating Wind Drift in Migrating Birds

Martin Greennw andThomas Alerstamw

wDepartment of Animal Ecology, Lund University, Ecology Building, SE-223 62 Lund, Sweden

(Received on 27 September 2001, Accepted in revised form on 24 May 2002)

Whether migrating birds compensate for wind drift or not is a fundamental question in birdmigration research. The procedures to demonstrate and quantitatively estimate wind drift orcompensation are fraught with difficulties and pitfalls. In this paper, we evaluate fourmethods that have been used in several studies over the past decades. We evaluate themethods by analysing a model migratory movement with a realistic scatter in flightdirections, for the ideal cases of full drift and complete compensation. Results obtained withthe different methods are then compared with the ‘‘true behaviour’’ of the model movement,illustrating that spurious patterns of drift and compensation arise in some cases. We alsoillustrate and evaluate the different methods of estimating drift for a real case, based ontracking radar measurements of bird migration in relation to winds. Calculating the linearregression of mean geographic track (resulting flight direction) and heading directions(directions of the birds’ body axis) of a migratory movement under different wind conditionsin relation to the angle a (the angle between mean track and heading) always provides robustand reliable results. Comparing mean flight directions between occasions with winds from theleft and right of the mean flight direction of the whole migratory movement also alwaysprovides expected and correct measures of drift. In contrast, regressions of individual flightdirections in relation to a (the angle between track and heading for the specific individuals orflocks) are liable to produce biased and spurious results, overestimating compensation/overcompensation if following winds dominate in the analysis and overestimating drift/overdrift if opposed winds are dominating. Comparing mean directions for cases with windsfrom the left and right in relation to individual flight directions also gives biased and spuriousresults unless there is full variation in wind directions or an equal distribution of crosswindsfrom left and right. The results of the methodological evaluation and the analysis of the realcase indicate that some earlier analyses of wind drift may have to be re-evaluated.

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1. Introduction

The resulting path over the ground of a flyingbird depends not only on the bird’s own flightdirection (direction of body axis) and speed butalso on wind direction and speed. If and to whatextent birds in flight are subjected to drift bycrosswinds is a fundamental question in bird

*Corresponding author. Tel.: +46 46 222 38 16; fax:46 46 222 47 16.E-mail address: [email protected] (M. Green).

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migration research, that has been addressed innumerous studies (cf. reviews by Alerstam, 1976;Richardson, 1991). Different studies have re-ported widely different answers, ranging fromcases where the migrants seem to be subjected tofull or partial wind drift, to cases where theycompensate completely for wind drift, or evenovercompensate.

This variability of birds’ behaviour in relationto winds in different situations is probably theresult of constraints in their capacity of wind

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M. GREEN AND T. ALERSTAM486

compensation under certain conditions as well asof adaptive variation in their responses to wind.When flying over moving surfaces, like sea orclouds, birds will not be able to prevent winddrift by orienting themselves with respect to afixed landscape and will therefore be subjectedto partial (over the sea) or full drift (Alerstam &Pettersson, 1976). Also flight at high altitude,particularly in darkness but perhaps also duringthe day, probably detracts from the birds’capacity of achieving full compensation for winddrift (Richardson, 1991). There is of course alsoa limit with respect to strong wind speed wherebirds will be overpowered by the winds andcannot avoid becoming drifted off their pre-ferred track direction (Alerstam & Hedenstrom,1998).

Depending on the wind pattern along theroute, the variability or constancy of windsbetween different flight steps, the distance to thedestination and differences between winds athigh and low altitudes, one can predict drift,partial drift, compensation or overcompensationto be adaptive or optimal in specific situations(e.g. Alerstam, 1979a, b, 2000, Stoddard et al.,1983, Liechti, 1995). Given the great importanceof wind for the economy and precision of birdmigration, it seems reasonable to assume thatbirds have evolved a high degree of flexibility intheir orientation response to wind. A numberof studies provide supporting evidence for suchadaptive drift or compensation behaviours(Richardson, 1991).

While birds’ responses to wind is such amultifaceted and challenging research question,the procedures to demonstrate and quantita-tively estimate wind drift or compensation inmigrating birds is fraught with difficulties andpitfalls. The average direction of a migratorymovement with a certain angular scatter may beshifted to the right with winds from the left andto the left with winds from the right as a result ofdifferential departures of migrants with differentpreferred track directions (probability of depar-ture increasing with potential wind assistancealong the intended track direction) even if thebirds in the migratory movement compensatecompletely for wind drift. This effect of ‘‘pseu-dodrift’’ was pointed out and analysed by Evans(1966) and Nisbet & Drury (1967), and graphi-

cally illustrated by Alerstam (1978). A quantita-tive estimate of pseudodrift requires informationabout (1) the frequency distribution of differenttrack directions in the migratory movementunder calm (no wind) conditions and (2) therelationship between probability of departureand wind (or expected ground speed; Nisbet &Drury 1967; Alerstam, 1978).

In this contribution, we will demonstrate thatspurious patterns of drift and compensation mayarise also for other purely trigonometric reasons,when analysing flight directions in relation towinds for migratory movements with a signifi-cant angular scatter. We will investigate poten-tial biases of different methods by makingcalculations for ideal cases of full drift andcomplete compensation. We will also illustrateand evaluate the different methods of estimatingdrift for a real case, based on tracking radarmeasurements of bird migration in relation towinds. By our evaluation, we intend to identifythe most robust and reliable methods ofestimating drift.

2. Triangle of Velocities

A flying bird’s movement over ground, itstrack direction and ground speed (the trackvector T), is the vector sum of its headingdirection (direction of body axis) and speedrelative the surrounding air (heading vector H),and the wind direction and speed (wind vectorW) according to the triangle of velocities shownin Fig. 1. Depending on whether the bird fliesalong a constant heading, constant track orsome intermediate situation in relation to thedirection towards its destination (D) we candefine three basic cases of (1) full wind drift[Fig. 1(a)], (2) complete compensation for winddrift [Fig. 1(b)] and (3) partial wind drift/partialcompensation [Fig. 1(c)], respectively. The anglebetween track and direction of destination, F,is the angle of drift. The angle between headingand direction of destination, y, is the angle ofcompensation. As seen from Fig. 1, the anglebetween track and heading, a, is in the drift caseequal to the angle of drift (F), in the compensa-tion case a ¼ y (the angle of compensation), andin the partial drift case a ¼ Fþ y: The angles dand b denote the angles between heading and

Fig. 1. The triangle of velocities for three differentsituations: (a) full wind drift, (b) complete compensationand (c) partial drift/partial compensation. T¼ track vector,the bird’s flight direction and speed over the ground,H¼ heading vector, the bird’s flight direction and speedrelative to the air, W¼wind vector, wind direction andspeed, D¼ direction of migration destination, a¼ anglebetween track and heading, b¼ angle between track andwind direction, d¼ angle between heading and winddirection, F¼ angle between track and destination direc-tion, i.e. angle of drift, y¼ angle between heading anddestination directions, i.e. angle of compensation.

WIND DRIFT IN MIGRATING BIRDS 487

wind direction and track and wind direction,respectively. Additional cases, not shown inFig. 1 and less frequently observed amongmigrating birds, are those of overcompensation,when the bird changes its heading more into thewind than necessary for full compensation (cf.Alerstam, 1979b), and overdrift, when the head-ing is adjusted to some degree along the winddirection, giving a tendency of downwindorientation. The analysis of drift or compensa-tion is based on the variation of track andheading directions in relation to wind and doesnot require the direction towards the destination(D in Fig. 1) to be specified. However, it isimplicitly assumed that D corresponds to theflight direction with zero wind (and mean Dfor a migratory movement corresponds to theintercept with a¼ 0 according to the regressionanalyses described below).

The nomenculture used is shown in AppendixA. In our calculations and figures, we haveassumed a fixed airspeed under all wind condi-

tions. This assumption is not strictly representa-tive of true bird behaviour during migrationas birds are expected to adjust airspeed to wind,and they have been shown to do so at leastroughly in several cases (cf. Pennycuick, 1978;Liechti et al., 1994; Liechti, 1995; Hedenstrom& Alerstam, 1995; Green & Alerstam, 2000).However, the adjustments in airspeed are rela-tively small and our approximation to a constantairspeed will affect the results of our calculationsonly to a minor degree and will have no influenceon our general conclusions.

If birds travel along fixed headings (full drift)with a constant airspeed, track will vary as a indifferent wind conditions according to

a ¼ arctana sin d

1 þ a cos d

� �: ð1Þ

If birds travel along fixed tracks (completecompensation) with constant airspeeds, headingswill vary as a in relation to winds according to:

a ¼ arcsinða sin bÞ: ð2Þ

where a is the ratio between wind speed and thebirds’ airspeed (Alerstam, 1976).

A practical difficulty associated with theanalysis of the triangle of velocities is thatthe accuracy differs between the measures ofthe track, wind and heading vectors, based onfor example radar studies of bird migration. Theheading vector will be most liable to errorsbecause it is calculated from the primarymeasurements of the track and wind vectors.Although such errors will cause noise in the datasets making it important to have large samplesizes, they are not likely to cause systematicbiases in the analyses of drift/compensation, andwe have not considered the effect of suchmeasurement errors in the present evaluation.

3. Methods

Several methods of analysing drift or com-pensation in migrating birds have been employedduring the years (for examples, see, Alerstam,1976; Richardson, 1979; Liechti, 1993; Alerstam& Gudmundsson, 1999a, b). We have investi-gated four different methods of estimating drift:


(1) The mean geographic track and headingdirections of a migratory movement for occa-sions with different wind conditions areregressed on the angle a (Fig. 1). With aregression slope of b (btrack hereafter) for therelationship between track direction and a, thecorresponding slope for heading direction inrelation to a (bhead hereafter) will always be(btrack � 1) because a is the angle between trackand heading direction (a¼ track minus headingdirection, which is positive when track directionis to the right of heading direction and negativewith track direction to the left of headingdirection). With complete compensation, trackdirection will remain constant irrespective ofwind and thus also of a, and the regression slopebtrack ¼ 0: For the same case, heading directionwill change in relation to a with a regressionslope bhead ¼ �1: With full drift, track directionwill change in relation to a with regression slopebtrack ¼ 1; while heading direction remains con-stant with bhead¼ 0. For intermediate casesof partial drift, btrack will fall between 0 and 1(btracko0 indicates overcompensation andbtrack41 overdrift, respectively). This means thatbtrack constitutes a quantitative measure of driftdirectly reflecting the relative magnitude of thedrift angle (f) in relation to the combined anglesof drift and compensation (a ¼ fþ y; cf. Fig. 1):

btrack ¼f

fþ y: ð3Þ

Confidence intervals of btrack can be calculatedby standard statistical linear regression methods.This measure of drift was originally proposedand used for analysing average directions ofmigratory movements recorded mainly by sur-veillance radar during different wind situations(Alerstam, 1976). The intercepts (with a ¼ 0) ofthe above-mentioned regressions correspond tothe assumed mean directions towards the mi-grants’ destinations (D in Fig. 1).

(2) The variation in track and headingdirections of flocks or single birds, belonging toa given migratory movement, are analysed inrelation to a according to the regression methoddescribed above. The difference between thismethod and method 1 is that the latter is basedon average directions for the whole migratory

movement while this method is based onindividual directions recorded for flocks or singlebirds. This type of data can be obtained fromstudies by tracking radar or optical trackinginstruments (Piersma et al., 1990; Alerstam &Gudmundsson 1999a, b; Green, 2001), but cansometimes also be extracted in studies usingsurveillance radar (Gudmundsson, 1994).

(3) Mean track and heading directions of amigratory movement are compared for situa-tions with winds from different sectors. Windsfrom the left and right, respectively, of the meanoverall direction will be associated with distinctdifferences in track or heading directions, orboth, depending on if drift, compensation orpartial drift prevails. Differences in directionsbetween different wind conditions may be testedaccording to circular statistics (Batschelet, 1981).This method gives a qualitative rather thanquantitative answer about drift/compensation.A calculation of the magnitude of drift canhowever be made as follows. If track directionsin winds from the left and right are denoted T1

and T2 and the corresponding heading directionsare denoted H1 and H2; then a1 will be T1 � H1

and a2 will be T2 � H2: The estimated magnitudeof drift is then

btrack ¼T1 � T2

a1 � a2

� �: ð4Þ

Calculations of confidence intervals for btrack arehowever not possible in this case.

(4) Mean track and heading directions offlocks or single birds in the migratory movementare compared between situations with windsfrom the left and right, respectively, of theindividual track direction. Hence, according tothis method it is the wind direction in relationto the individual track directions and not to theoverall mean direction (as in the precedingmethod) that is the basis of subdivision intodifferent wind situations. Estimates of themagnitude of drift can be made as describedabove [eqn 4].

All of the above methods give correct results ifthere is no or exceedingly small scatter of trackdirections in the migratory movement undercalm conditions. However, such migratorymovements are rarely met with, and usually the

Fig. 2. A graphical illustration of the calculations oftracks and headings in (a) the drift case and (b) thecompensation case. A circle with the radius equal to thebird’s heading vector (broken lines) is drawn from the topof the wind vector (arrow). The resulting track vectors canbe drawn from the base of the wind vector as shown by theunbroken lines. In our model migratory movement we haveassumed a constant airspeed, a wind speed of 0.5 timesthe bird’s airspeed, fixed headings (a) or tracks (b) in 101intervals 7501 around a mean towards 901 (due east). Inthis particular example, we show the resulting tracks andheadings in a following wind differing 301 from the meanheading (a) or track (b), i.e. a wind direction of 2401. Theangle a (the angle between track and heading) is shown forthe outermost and central cases.


analysis must deal with migratory movementscomposed of different populations and/or spe-cies with different destinations and thus with aconsiderable scatter of flight directions. It is thisscatter that causes the problems of spuriousresults for some of the methods (see below).

To evaluate these problems we have assumeda migratory movement with a uniform spread ofdirections 7501 around the mean directionunder calm conditions. This corresponds to arealistic scatter (mean vector length r ¼ 0:85;angular deviation¼ 311; cf. Batschelet, 1981) formany cohorts of migrants encountered in radarand field studies. Of course, real migratorymovements do not show a uniform frequencydistribution of directions, but our model exam-ple will serve the purpose of clearly demonstrat-ing the biases associated with different methodsof analysis (see Section 5 for the analysis of a realcase). Assuming (a) full drift and (b) completecompensation for birds in the model movement,we solved the triangle of velocities [eqns (1) and(2)] for each 101 of flight directions (n ¼ 11calculations for the total 1001 spread of themigratory movement) as illustrated in Fig. 2.Wind speed was kept constant at 0.5 times thebirds’ airspeed. We made calculations for each101 of wind directions around the full circle,corresponding to a total of 36 different windsituations. With 11 flight directions in 36 windsituations, the total number of trigonometriccalculations was 396 for each of the two cases ofdrift (a) and compensation (b), respectively. Wethen analysed the total sample and subsamplesof these data by the four above-mentionedmethods in order to investigate to what degreethey are liable to produce biased or spuriousresults rather than the correct answers of fulldrift or complete compensation, respectively.

4. Results: Estimating Drift for ModelMovements with Known Patterns of Drift

and Compensation

4.1. REGRESSIONS OF MEAN TRACK AND HEADING

DIRECTIONS IN RELATION TO a

Plots of average geographic track and headingdirections in relation to a for our modelmigratory movement (cf. above) under differentwind situations are shown in Fig. 3. Here we

Fig. 3. Plots of mean track (open symbols) and heading directions (filled symbols) in relation to a (the angle betweentrack and heading) for the model movement under different wind situations. Here we have defined each wind situation aswinds from a 101 sector. The case of full drift is illustrated in (a) and that of complete compensation in (b). btrack and bheading

refer to the regression slopes of track and heading direction in relation to a (cf. text).


used each wind direction (101 interval) as aspecific wind situation. This analysis gives as aresult the expected and correct regressions withbtrack ¼ 1 and bhead ¼ 0 in the drift case[Fig. 3(a)] and btrack ¼ 0 and bhead ¼ �1 in thecompensation case [Fig. 3(b)]. We also calcu-lated the regressions for average geographictrack and heading directions in relation to afor six larger wind sectors (wind situations) on a,calculating mean directions for tailwinds fromthe left-hand side and from the right-hand side(451 sectors), crosswinds from left and right (901sectors) and headwinds from left and right (451sectors). Also, these regressions give the samecorrect results as above. It should also be noticedthat full variation in wind directions, i.e. windsfrom all directions (3601), is not required toproduce correct results with this method. Evendata from a small set of wind situations, e.g. onlywinds from the tailwind sector, are possible toanalyse and will produce correct results as long

as there are different wind situations within thetailwind sector that allow computation of reli-able mean directions. In the extreme case with novariation in winds, i.e. a constant wind directionin the whole material, there is of course nopossibility to analyse if the birds are drifted ornot (compare with method 2 below).

4.2. REGRESSIONS OF INDIVIDUAL TRACK AND

HEADING DIRECTIONS IN RELATION TO a

Regressions of geographic flight directions(tracks and headings) of individuals or flocks inrelation to a (not shown) produce expected andcorrect results (btrack ¼ 1 and bhead ¼ 0 in thedrift case and btrack ¼ 0 and bhead ¼ �1 in thecompensation case), only if there is full variationin wind situations, i.e. a uniform distribution ofwinds from all different sectors around the full3601. This situation is however seldom met within the field as migratory intensity is strongly


correlated with following winds (Richardson,1978, 1990), making it likely that winds from arestricted sector dominate in analyses of realmigratory movements.

With a non-uniform distribution of winds, theresults by this method of analysis will be biased.This is most easily seen by considering ananalysis for one specific wind situation. Suchan analysis is possible to conduct by thismethod, although it is non-sensical from thepoint of view of testing drift or compensation(the existence of wind variation will of coursealways be a pre-requisite for such a test).

Plots of individual tracks and headings inrelation to a for specific wind situations areshown in Fig. 4. In this figure, we show thepatterns in due tailwinds, due headwinds, cross-winds from the right-hand side and crosswindsfrom the left hand side for the model migratorymovement in full drift [Fig. 4(a–b)] and forfull compensation [Fig. 4(c–d)], respectively.The analyses show completely unexpected (andfalse) patterns, differing greatly from the ex-pected relationships of btrack ¼ 1 and bhead ¼ 0 inthe drift case and btrack ¼ 0 and bhead ¼ �1 in thecompensation case. Depending on the winddirection, the plots indicate spurious patternsof drift or overdrift (positive btrack) or over-compensation (negative btrack) irrespective of thebirds’ actual behaviour. An analysis includingseveral wind situations will indicate the truepatterns of drift or compensation only if there isa symmetrical balance of wind situations (con-sider the combination of some of all situationsin Fig. 4). The large scatter and risk of bias willhowever make such tests based on a combina-tion of wind situations of little use.

A more realistic case is shown in Fig. 5. Here,we have calculated regressions of tracks andheadings in relation to a for all individual datapoints in the model movement within thetailwind sector (defined as due tailwind inrelation to the average track and heading incalm situations 7451). Again, false patternsemerge indicating overcompensation (btracko0;bheado� 1) for the case when the birds aredrifted [Fig. 5(a)] as well as when they compen-sate completely [Fig. 5(b)]. Making the sameanalysis for winds from the headwind sector(not shown) indicates spurious patterns of drift

(btrack40), for both drifting and compensatingbirds. Considering winds from the crosswindsectors (not shown) indicates spurious over-compensation patterns for drifting birds andgives inconclusive patterns for compensatingbirds.

4.3. COMPARISONS OF MEAN TRACK AND HEADING

DIRECTIONS BETWEEN SITUATIONS WITH WINDS

FROM THE LEFT AND RIGHT

Comparing mean flight directions in windsfrom the left- and right-hand side in relation tothe mean flight direction of the whole migratorymovement always provides expected and correctmeasures of drift. If track direction changes withthe wind there is drift, and if track directionremains constant there is compensation. Evenwith a small scatter in wind directions, as whenthe whole material consists of data collected inwinds from the tailwind sector (or any othersector), this method gives correct results. All thatis required is that there is sufficient variation inwinds to make a subdivision of the material intowinds from the right and left, respectively.

4.4. COMPARISONS OF INDIVIDUAL TRACK AND

HEADING DIRECTIONS BETWEEN SITUATIONS

WITH WINDS FROM THE LEFT AND RIGHT

Dividing the data set into cases with windsfrom the left and right on the basis of flightdirections of individuals and/or flocks, andcomparing the mean directions of these groupsmay give biased and spurious results unless thereis full variation in wind directions or an equaldistribution of crosswinds from both left andright. If tailwinds are dominating, this methodproduces false indications of overcompensation,and if headwinds are dominating of drift/over-drift, irrespective of true bird behaviour, exactlyin the same way as the regression method 2above.

5. Estimating Drift for a Real MigratoryMovement: a Case Study

We will now use the four methods above in ananalysis of drift/compensation behaviour on areal field material of migrating birds. The dataset consists of arctic birds (mainly waders andskuas), leaving the Siberian tundra in easterly

Fig. 4. Plots of individual track and heading directions in relation to a (the angle between track and heading) for themodel movement in four different wind situations: due tailwinds (open squares), headwinds (filled squares), crosswinds fromthe left (filled triangles) and crosswinds from the right (open triangles). (a) and (b) refer to the case of full drift in (a) and tothat of complete compensation in (b).


Fig. 5. Linear regressions of individual tracks and headings in relation to a (the angle between track and heading) for allbirds (11 different directions) in the model migratory movement in tailwinds (winds blowing towards 7401 of the meanmigratory direction, nine different wind directions). Filled symbols, solid lines¼ tracks, open symbols, brokenlines¼ headings. (a) refer to the case of full drift and (b) to that of complete compensation.


directions (tracks: mean 10317211, r ¼ 0:93;headings: mean 10517251, r ¼ 0:91; N ¼ 569)in late summer on their way towards intermedi-ate staging or wintering areas in the Pacificregion. The material was collected by ship-bornetracking radar in the Arctic Ocean (Alerstam &Gudmundsson, 1999a). The data set was ana-lysed by Alerstam & Gudmundsson (1999a, b)according to methods 2 and 4, i.e. individualtrack and heading directions were regressed on aand mean flight directions in winds from leftand right of the individual birds’ flight directionswere compared. The result of both methodsindicated complete compensation for wind drift(Alerstam & Gudmundsson, 1999a, b). Sincemost trackings were made in tailwind situations,there is a risk of obtaining spurious results withthese methods (see above).

In Table 1, we compare the results of thedifferent methods. In Fig. 6, we illustrate the

distribution of track and heading directions inwinds from the right- and left-hand side inrelation to the average track direction in calmsituations (tracks: mean 9817161, r ¼ 0:96;headings: mean 10117151, r ¼ 0:96; N ¼ 54;not significantly different from the mean direc-tions in the whole material, Watson–Williamstest, as ‘‘calm situations’’ we used all trackingsmade in wind speeds below 3m s�1). While theearlier analysis (methods 2 and 4) showedcomplete compensation for drift, the othermethods (methods 1 and 3) showed a highlysignificant drift effect in the order of full drift(Table 1). In light of our methodologicalevaluation above, we thus conclude that therewas in fact a significant drift effect for thismigratory movement. However, it still remainsto be shown whether this drift effect could beattributed to true drift or pseudodrift (sensu

Evans, 1966; Nisbet & Drury, 1967).

Table 1Results of drift analysis of migrating birds leaving the Siberian tundra on autumn migration

Track Heading

Method N btrack CI of btrack Level ofsignificance

bhead CI of bhead Level ofsignificance

Conclusion

1a 6 0.94 0.57 / 1.31 ** �0.06 �0.43 / 0.31 NS Full drift1b 13 0.67 0.23 / 1.11 *** �0.33 �0.77 / 0.11 NS Partial drift/full drift2 569 0.05 �0.08 / 0.17 NS �0.95 �1.08 / -0.83 *** Full compensation3 569 (0.94) *** (�0.06) NS Full drift4 569 (0.12) NS (�0.88) *** Full compensation

Note: The material has been analysed with four different methods (see text). 1a¼Linear regression of mean track andheading directions in six large wind sectors in relation to a (see text), 1b¼ linear regression of mean track and headingdirections in 13 small wind sectors (101) in relation to a , 2¼ linear regression of individual track and heading directions inrelation to a, 3¼ comparison of mean flight directions in winds from right and left in relation to average migratory direction,4¼ comparison of flight directions of individuals/flocks in winds from right and left. The drift and compensation coefficient,btrack and bhead ; are calculated as the slopes of the regression lines (methods 1 and 2) and estimated according to eqn (4) formethods 3 and 4 (given within parentheses). Statistical testing have been made according to conventional linear regressionmethods for methods 1–2 and with circular statistical methods (Batschelet, 1981) for methods 3 and 4. Differences indirections with method 3 were tested with the Watson–Williams test, differences with method 4 were tested with a chi-squaretest. CI¼ 95% confidence intervals for the regression coefficients. Test methods 3 and 4 involve the analysis of two groupswith a combined total sample size of 569 cases.


6. Conclusions

Our conclusion from the evaluation of meth-ods above is that both linear regression of meanflight directions for whole migratory movementsin different wind situations in relation to a(method 1) and comparisons of mean flightdirections in different winds in relation to themean direction of the whole migratory move-ment (method 3), produce robust and reliableresults in all situations. Thus, these methodsshould be used to evaluate whether birds driftwith the wind or compensate for drift. Theregression method gives a direct measure of themagnitude of the drift (btrack) and confidenceintervals for btrack can be computed by standardstatistical methods. Also, method 3 gives thepossibility to estimate a corresponding measureof drift as the regression method (see above, eqn(4)), although this method is primarily qualita-tive (indicating if there is a significant drift effector not). Methods 2 and 4 should be avoided asthey may give rise to spurious results andmisinterpretations of the data. In these cases,the outcome of the analysis depends strongly onthe distribution of winds, showing reasonableresults only if there is a uniform/symmetricdistribution of winds from different directions.The reason for the failure of methods 2 and 4 is

that the between-individual/flock differencesin track and heading directions confound theanalyses when it is assumed that responses ofdrift or compensation by the birds constitute theprimary source of variation in flight directions.More specifically, with following winds, trackdirections will always be more concentrated thanheading directions in the migratory movement,and under such winds, methods 2 and 4 will givefalse implications of compensation/overcompen-sation. With headwinds, heading directions willalways be less scattered than track directions inthe migratory movement, and when such windsdominate in the analysis, methods 2 and 4 willgive false indications of drift/overdrift (cf. Fig. 4when also the misleading patterns associatedwith crosswinds are shown).

It should be remembered that if a criticalanalysis demonstrates the existence of a signifi-cant drift effect, it still remains to be consideredif pseudodrift sensu Evans (1966) and Nisbet andDrury (1967) may provide the full explanation,or if one can draw the conclusion that the birdsare subjected to true wind drift (full or partial) inthe particular case analysed. A clearer under-standing of constraints and adaptations in birds’orientation responses to wind requires furtherstudies and analyses under a variety of condi-tions, with due consideration of the difficulties

Fig. 6. Distribution of tracks (a) and headings (b) ofradar-tracked arctic migrants, with easterly (0-180o) flightdirections on the Siberian tundra [see Alerstam &Gudmundsson (1999a) for details]. Bars show percentagedistribution in 10o intervals. Arrows show mean directions.Filled bars and solid arrows show birds in winds from theright (south) and open bars and broken arrows show birdsin winds from the left (north).


and biases involved in estimating drift/compen-sation for migrating birds.

We hope that this methodological evaluationwill give rise to more studies of whether birdsdrift or compensate, and perhaps a re-evaluationof some older studies, and thus to a betterunderstanding of birds’ capacities and adapta-tions in their orientation relative to wind.

We are grateful to Gudmundur A. Gudmundssonfor thought provoking discussions, and to twoanonymous referees for valuable comments. We alsothank the Swedish Polar Research Secretariat fororganizing the expeditions where we obtained theempirical data analysed in this paper. This study was

supported by the Swedish Natural Science ResearchCouncil.

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Appendix A

Nomenclature

D direction of destination of migrationH heading vector and heading direction (flight direction along the bird’s body axis)T track vector and track direction (resulting flight direction over the ground)W wind vector and wind directiona the ratio of wind speed to the bird’s air speed (speed relative to the surrounding air)btrack drift coefficient, the slope of the regression of track direction on abhead the regression slope of heading direction on aa the angle between track and heading direction, angle of drift in case of full drift, angle

of compensation in case of complete compensationb the angle between track and wind directiond the angle between heading and wind directiony the angle between heading direction and direction of destination, angle of compensationF the angle between track direction and direction of destination, angle of drift

the problem of estimating wind drift in migrating birds

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