The Effect of Animal Movement on Line Transect Estimates of Abundance

Abstract

Line transect sampling is a distance sampling method for estimating the abundance of wild animal populations. One key assumption of this method is that all animals are detected at their initial location. Animal movement independent of the transect and observer can thus cause substantial bias. We present an analytic expression for this bias when detection within the transect is certain (strip transect sampling) and use simulation to quantify bias when detection falls off with distance from the line (line transect sampling). We also explore the non-linear relationship between bias, detection, and animal movement by varying detectability and movement type. We consider animals that move in randomly orientated straight lines, which provides an upper bound on bias, and animals that are constrained to a home range of random radius. We find that bias is reduced when animal movement is constrained, and bias is considerably smaller in line transect sampling than strip transect sampling provided that mean animal speed is less than observer speed. By contrast, when mean animal speed exceeds observer speed the bias in line transect sampling becomes comparable with, and may exceed, that of strip transect sampling. Bias from independent animal movement is reduced by the observer searching further perpendicular to the transect, searching a shorter distance ahead and by ignoring animals that may overtake the observer from behind. However, when animals move in response to the observer, the standard practice of searching further ahead should continue as the bias from responsive movement is often greater than that from independent movement.

Full text

RESEARCH ARTICLE
The Effect of Animal Movement on Line
Transect Estimates of Abundance
Richard Glennie*, Stephen T. Buckland, Len Thomas
Centre for Research into Ecological and Environmental Modelling, University of St Andrews, St Andrews, UK
*rg374@st-andrews.ac.uk
Abstract
Line transect sampling is a distance sampling method for estimating the abundance of wild
animal populations. One key assumption of this method is that all animals are detected at
their initial location. Animal movement independent of the transect and observer can thus
cause substantial bias. We present an analytic expression for this bias when detection with-
in the transect is certain (strip transect sampling) and use simulation to quantify bias when
detection falls off with distance from the line (line transect sampling). We also explore the
non-linear relationship between bias, detection, and animal movement by varying detect-
ability and movement type. We consider animals that move in randomly orientated straight
lines, which provides an upper bound on bias, and animals that are constrained to a home
range of random radius. We find that bias is reduced when animal movement is constrained,
and bias is considerably smaller in line transect sampling than strip transect sampling pro-
vided that mean animal speed is less than observer speed. By contrast, when mean animal
speed exceeds observer speed the bias in line transect sampling becomes comparable
with, and may exceed, that of strip transect sampling. Bias from independent animal move-
ment is reduced by the observer searching further perpendicular to the transect, searching
a shorter distance ahead and by ignoring animals that may overtake the observer from be-
hind. However, when animals move in response to the observer, the standard practice of
searching further ahead should continue as the bias from responsive movement is often
greater than that from independent movement.
Introduction
Line transect sampling is a distance sampling method [1,2], widely used for estimating the
abundance of wild animal populations. Lines are placed randomly across a study region, or
more usually a systematic grid of lines is randomly positioned, and the observer traverses each
line, recording the perpendicular distance from the line to each detected animal. Typically, the
observer fails to detect all animals in the surveyed strip, and these distances are used to model
the probability of detection, and hence to estimate abundance.
PLOS ONE | DOI:10.1371/journal.pone.0121333 March 23, 2015 1/15
a11111
OPEN ACCESS
Citation: Glennie R, Buckland ST, Thomas L (2015)
The Effect of Animal Movement on Line Transect
Estimates of Abundance. PLoS ONE 10(3):
e0121333. doi:10.1371/journal.pone.0121333
Academic Editor: Marco Festa-Bianchet, Université
de Sherbrooke, CANADA
Received: November 18, 2014
Accepted: February 10, 2015
Published: March 23, 2015
Copyright: © 2015 Glennie et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.
Data Availability Statement: All relevant data are
within the paper.
Funding: This work was supported by the University
of St Andrews (http://www.st-andrews.ac.uk/; RG,
STB, LT) and by a summer scholarship and PhD
grant from The Carnegie Trust for the Universities of
Scotland (http://www.carnegie-trust.org/) to RG. The
funders had no role in study design, data collection
and analysis, decision to publish, or preparation of
the manuscript.
Competing Interests: The authors have declared
that no competing interests exist.

The method relies on three key assumptions, and if these are not met then estimates of
abundance can have substantial bias [1] (pp. 29–37). These are:
1. animals on the line are detected with certainty;
2. measurements are exact;
3. animals are detected at their initial location.
When the first assumption is not met, double-observer (or multi-observer) methods, which
combine distance sampling with mark-recapture, may be used [3,4]. Similarly, if the second as-
sumption is violated, measurement error models [5,6] can be adopted. We concentrate on vio-
lations of the third assumption.
Conceptually, line transect sampling is a snapshot method [1](pp.31),thatis,weconsiderthat
theanimalsareatafixedlocationwhilethesurveytakesplace.Thus,wehaveprobabilisticen-
counters between a moving observer and immobile animals [7]. There are models [8,9]foren-
counters between mobile animal populations and observers, but these are limited in use: they rely
upon quantities that are difficult to determine (mean animal speed, encounter radius) and they as-
sume an ideal free gas movement model which can be unrealistic [10]. Making the third assump-
tion avoids such problems. This assumption can be violated in two ways: animals can move in
response to the observer, or move independently of the observer. Responsive movement is a com-
mon problem in distance sampling surveys and the bias it causes can be reduced by modelling the
movement [11,12] or using double-observer methods [13]. In a few specific cases, independent
movement has been modelled (for fish: [14]; for seabirds: [15]) to reduce bias; however, these
methods are ad hoc and specific to their application. We do not explore responsive movement
here; instead, we consider how animal movement independent of the observer affects bias. For
whale surveys, it was concluded that such movement may seriously bias the estimate of mean den-
sity if the speed of the whales approaches that of the observer, if the encounter region is long rather
than wide, or if the probability of sighting a whale is not strongly dependent on the time it spends
within that region [16]. Subsequently, simulation revealed that bias was negligible if mean animal
speed was one quarter of that of the observer, but not if animal speed was one half that of the ob-
server [17]. This conclusion has now been adopted within distance sampling [1] (pp. 131, 173)
and is used to determine if independent animal movement is a problem in particular studies [18].
This is despite these results being pertinent only to the single simulation scenario considered.
In this paper, we quantify the bias caused by independent animal movement in some exam-
ple cases and discuss the non-linear relationship between movement, detection and bias. We
first consider strip transect sampling which is a special case of line transect sampling in which
all animals within a strip are assumed to be detected [1] (pp. 2–3). We derive an analytic ex-
pression for the bias caused when animals move in straight lines, each with the same constant
speed but an individual random direction, within a rectangular study area. We then generalize
to surveys in which detectability falls off with distance from the line, and for which animal
movement is more complex, using simulation to quantify bias. Finally, we discuss what these
results reveal about bias caused by animal movement independent of the observer.
Strip Transect Sampling
Consider estimating the abundance of an animal population using strip transect sampling
where the study area is rectangular of length aand breadth Lwith corners, in (x,y)-coordi-
nates, at (−a/2, 0), (a/2, 0), (−a/2, L), (a/2, L). We assume the number of animals within the
study area is fixed, and so adopt a wrap-around model: for each animal that exits across one
boundary of the study area, a new animal enters immediately at the same distance along the
The Effect of Animal Movement on Line Transect Estimates of Abundance
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opposite boundary. Intuitively, we think of the study area as part of a wider region. We assume
the (strip) transect is placed randomly within this region and so may take the transect to be at
x= 0. Let the transect have half-width w, where 2w<a, extending out to either side of x=0
and length L, so it transverses the entire study region in the y-direction.
We assume animal itravels in a straight line at speed uand in a direction θ
i
where θ
i
is a
random deviate from a uniform distribution U(0, 2π](Fig. 1). Thus, the equations of motion of
the ith animal are given by:
yi;t¼yi;0þucos ðyiÞt
xi;t¼xi;0þusin ðyiÞt
where tis time and (x
i,0
,y
i,0
) is the initial (t= 0) location of animal i.
The observer travels along the centerline, x= 0, at speed v>0, starting at y= 0. Thus, the
position of the observer at time tis given by (0, z
t
) where z
t
=vt. We define the encounter region
to be the area around the observer within which every animal is recorded. No animals are re-
corded outside the encounter region. Here, we assume the encounter region is rectangular, ex-
tending a distance weither side of the centerline and a distance l, which we call the lookahead
distance, in front of the observer (Fig. 1). The encounter region is wrapped using the wrap-
around model to fix its size. (There are practical methods of implementing a wrap-around de-
sign with strip transects, using buffer zones [19].)
When estimating abundance using strip transect sampling, it is assumed that the animals
are immobile—this is termed a snapshot method as the survey is assumed to take place instan-
taneously [1] (pp. 31). When true, the expected proportion of animals within the strip is
P1¼2w
a. However, if animals are mobile then they can move into the encounter region and so
be recorded; hence, we expect to record more animals, which causes positive bias in
abundance estimates.
Animals not initially inside the encounter region can enter in three ways: from in front,
from the side and from behind. We shall consider each in turn. First, bias is not caused by ani-
mals travelling toward the observer and entering the encounter region from in front since on
average in any interval of time an equal number of animals, that if immobile would be re-
corded, will move away from the observer and not be recorded. Second, bias is caused by ani-
mals entering the encounter region from the side. Any point along the strip is within the
encounter region of length lahead of the observer for l
vunits of time. In that time, all animals
with direction θand speed utravel lusinðyÞ
vunits of distance perpendicular to the line. Hence the
proportion of all such animals not initially inside the encounter region entering from the side
and so being recorded is given by lujsinðyÞj
av . Given that θis uniform on (0, 2π], the expectation of
this proportion is
P2¼luE½j sin ðyÞj
av ¼2lu
pav
as E½jsinðyÞj ¼ 2
p.
Lastly, animals can only enter the encounter region from behind by overtaking the observer;
hence, no bias is caused when animals travel slower than the observer. Under our movement
model, only animals that are initially behind the transect, y<0, can overtake the observer and
enter the encounter region. By the equations of motion given above, an animal with direction θ
and initial location (x
0
,y
0
) overtakes the observer at time
T¼y0
vucos ðyÞ
The Effect of Animal Movement on Line Transect Estimates of Abundance
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Thus, to have 0<T<L
vwhen y
0
<0 and have the animal recorded, we require that the follow-
ing inequalities are satisfied:
v
u<cos ðyÞ
b
vðucos ðyÞvÞy00
wxTþkw
for some k2[0, T
δ
] where Td¼l
ucosðyÞvis the time the animal is within the encounter region.
From these inequalities, we see that within the subset of those animals with direction θsuch
Fig 1. Strip transect with encounter region. Transect line (dashed grey line) with the strip extending a
distance wout to either side (solid grey lines). The observer and encounter region at time t−1 (dashed black)
and t(solid black) are shown, together with an animal moving with speed uin direction θ
doi:10.1371/journal.pone.0121333.g001
The Effect of Animal Movement on Line Transect Estimates of Abundance
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that cosðyÞ>v
u, the proportion that are recorded is given by
ð2wþujsin ðyÞjTdÞL
vðucos ðyÞvÞ

aL
The expected value of this proportion with respect to θis
2wffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2v2
p
cos 1v
u

av 2w
aþ
ul 1v
u

av cos 1v
u

since EjsinðyÞj jcosðyÞ>v
u

¼1v
u
cos1ðv
uÞand EcosðyÞjcosðyÞ>v
u

¼ffiffiffiffiffiffiffiffi
u2v2
p
ucos1v
u
ðÞ
. Finally, since the
proportion of animals with required direction θis given by PcosðyÞ>v
u

¼1
pcos1v
u

, the
total proportion of animals causing this bias is
P3¼2wffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2v2
p
apv2w
pacos 1v
u

þlðuvÞ
pav
We are now able to give an expression for the total bias in the abundance estimate. If Pis
the total expected proportion of animals detected in the strip transect, then when uv,P=P
1
+P
2
and when u>v,P=P
1
+P
2
+P
3
. Thus, strip transect sampling produces an abundance
estimate with percentage bias given by
100 
ul
pwv uv
ul
pwv þffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2v2
p
pv1
pcos 1v
u

þluvðÞ
2wpvu>v
8
>
>
>
<
>
>
>
:
In particular, if movement is absent (u= 0) or animal speed is less than observer speed (uv)
and the observer only searches abeam, not ahead (l= 0), then there is no bias. Nevertheless, the
bias caused can be substantial. For example, if we assume that the observer searches as far
ahead as to the side, l=w, and the animals are slower than the observer, u<v, then bias is the
ratio of animal speed to observer speed, divided by π. Thus, if animal speed is one third observ-
er speed, bias is 1/(3π) = 10.6%.
Line Transect Sampling
In line transect sampling, some animals in the covered strip may be undetected. Movement in-
dependent of the observer causes bias in the estimated probability of detection, and this bias
combines with the bias arising from movement of animals into the encounter region, so that
the formulae of the previous section no longer apply.
The detection process can be modelled as either a continuous [20,21] or a discrete [20,22]
hazard-rate process. The discrete model is more appropriate for surveys of whales, when detec-
tion cue is primarily a whale blow, or for songbirds, if the cue is a songburst. Hiby adopted a
discrete hazard-rate model [17]. To avoid introducing another parameter (cue rate), we adopt
a continuous-time model.
By combining animal movement with a hazard-rate detection process, the problem of deriv-
ing an analytic expression for bias appears intractable as it is a non-linear function of the mean
probability of detection. Therefore, a simulation study was performed in order to quantify bias
The Effect of Animal Movement on Line Transect Estimates of Abundance
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and investigate the relationship between bias, movement and detection. We first describe the
simulation setup (Section 1) and then present and discuss the results as they relate to bias (Sec-
tion 2), the detection function (Section 2.1), lookahead distance (Section 2.2) and truncation
distance (Section 2.3).
1 Simulation Setup
We consider two animal movement types at a range of speeds relative to the observer. We also
vary the maximal distance the observer searches ahead (the lookahead distance) and the shape
of the detection function: both control the size of the encounter region around the observer.
The study area and transect are defined similarly to the strip transect case given above. The
study area was one square kilometre and a wrap-around model was used. The transect was one
kilometre long and bisected the study area. There were 100 animals distributed, for each simu-
lation, randomly across the study area.
Two types of animal movement were considered. First, the linear movement model, the
same as in the strip transect case, was used, where all animals moved in straight lines at the
same speed but each in a randomly assigned direction. It is particularly useful to quantify the
bias arising from this model: for any other random animal movement, each animal will travel
through a smaller range of perpendicular distances from the line compared with linear move-
ment, and so bias will be less as bias depends critically upon how far an animal travels from its
initial location. Thus, the bias arising from linear movement is an upper bound for the bias
caused by any other type of random animal movement. The second model considered is a
home range model: each animal is contained within a circular home range with a randomly as-
signed radius from a Gamma distribution with mean 20 and variance 11. Inside its home
range, the animal performed a correlated random walk, where the direction of travel was per-
turbed at each time step of the simulation by adding a random deviate from a wrapped normal
with zero mean and 0.1 variance. When an animal reached the boundary of its home range, it
was reflected back by changing its direction of travel by πradians. By restricting the range of
perpendicular distances from the line the animal may travel through, we expect the home
range model to produce smaller bias than the linear movement model; clearly, however, this
will depend critically on the radius of the home ranges. If home range radius greatly exceeds
the strip half-width w, then the constraints to movement would have little effect on the bias. If,
however, the radius is smaller than w, as might occur with many songbird territories, then the
constraints on movement should result in appreciably lower bias. For each movement model
we consider animal speeds from 0 m s
−1
to 20 m s
−1
where all animals have the same speed. If
each animal’s speed were to vary around a population mean, we expect bias would be similar
(as indicated by some preliminary simulation) since bias depends strongly on the mean speed
of the population rather than on an individual’s instantaneous speed.
The observer travels along the line at 10 m s
−1
and the probability of detecting an animal is
modelled as a continuous hazard-rate process. However, as simulation studies are, by nature,
discrete we integrate this hazard over the time step used in the simulation to give the probabili-
ty of being detected within that time step. Thus, the probability of being detected in time step t
with the continuous radial hazard k(r)=cr
−d
where c>0, d>2is
gtðr0;r1Þ¼1exp xt
s2

b
Ix2
t
r2
1
Ix2
t
r2
0
0
B
B
@
1
C
C
A
0
B
B
@
1
C
C
A
where x
t
is the perpendicular distance of the animal from the line at time t,r
0
is the radial
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distance of the animal from the observer at the beginning of the time step, r
1
is the radial dis-
tance at the end of the time step, b=d−1, sb¼cGð0:5bÞGð0:5Þ
2Gð0:5dÞand I
r
is the cumulative distribution
function of a Beta random variable evaluated at rwith parameters (b, 0.5). Here, we use a time
step of 0.05s duration. We will also consider different sizes of the encounter region by changing
the shape parameter (b= 2, 3, 5) and the lookahead distance (l= 0, 5, 10, 15, 20, 25, 30). Finally,
we will investigate the effect that truncating the data at the analysis stage has on bias by trun-
cating at distances where the true detection function appoximately equals 0.05, 0.1, 0.2, 0.4, 0.6
and 0.8.
For each animal speed and movement model, we simulated one thousand surveys consisting
of five hundred transects each. We did not consider any model selection: we fit a hazard-rate
model only. The simulated data were analysed using the Multiple-Covariate Distance Sampling
(MCDS) engine that is included in the software Distance [23]. The computer code used to per-
form this simulation study is included as online supporting material (S1 Code).
2 Simulation Results
Fig. 2 gives the bias estimated from the simulation study (with hazard-rate shape parameter
b= 2) for each relative animal speed and movement model. The bias caused in strip transect
sampling, calculated from the analytic expression in Section 1, with the observer searching as
Fig 2. Percentage bias in the abundance estimate for strip and line transects. Bias for strip transect
sampling (solid line) calculated from the analytic expression given and for line transects (hazard-rate b=2)by
averaging over 1000 simulations for linear movement (dashed line) and home range movement (dotted line)
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far ahead as to the side, is also included in the figure for comparison. The strip transect bias in-
creases linearly when animal speed is less than observer speed, and then increases nonlinearly
and rapidly thereafter. Bias in line transect sampling under both movement models behaves
similarly as animal speed increases, but the bias at each speed is less than in strip transects. For
strip transect sampling, bias reaches nearly 10% when animal speed is just 30% of observer
speed, while for our simulation of line transect sampling under linear animal movement, bias
does not reach 10% until animal speed is 80% of observer speed. However, for this scenario, at
very large animal speeds (150% –200% observer speed) the bias in line transect sampling is
similar to or exceeds that in strip transect sampling.
We now consider these results. First, the difference in bias between linear movement and
home range movement confirms that bias is related closely with the range of perpendicular dis-
tances from the line that an animal can travel through. At smaller speeds, the animals move
through a similar range of perpendicular distances under each movement model which ex-
plains why bias is similar between the two. However, at larger speeds, animals under the linear
movement model travel through a large range of perpendicular distances compared with the
home range model where animals are constrained to a local territory; hence, linear movement
causes more bias at these speeds. This shows that bias from animal movement is related to the
distance an animal of interest may travel—the smaller that distance, the less the bias. Second,
these results show that any difference in the bias in strip and line transect sampling is due to
the detection process as line transect sampling is a generalisation of strip transect sampling
where detection of animals is uncertain.
2.1 Detection Function. Fig. 3 shows that as speed increases, the estimated detection
probability falls off more rapidly and the shoulder (range of distances from the line for which
probability of detecting an animal is one) narrows. This can also be seen in the distribution of
observed distances in Fig. 4. The effect on bias of changing the shape parameter of the detection
function is shown in Fig. 5. Here, we discuss how the bias in the estimated detection function
occurs, how it relates to bias in abundance estimates, and how the shape parameter of the func-
tion affects bias.
Bias in the estimated detection function is a result of animal movement and the shape of the
true detection function. As speed increases, the number of detections close to the line increases
rapidly since more animals are able to travel very close to the line during the survey and so be
detected. The number of detections at distances further from the line does not increase as rap-
idly since some of the animals travelling toward the line that are detected replace those that re-
main undetected by travelling away from the line. This does not occur very near to the line
since there the probability of detection is close to one and so the animals travelling both toward
and away from the line are detected. The rapid fall of detections as distance from the line in-
creases is the reason the detection function falls more rapidly at greater speeds. This bias in the
estimated detection function generates positive bias in the abundance estimate as the probabili-
ty of detection is underestimated.
Nevertheless, bias in line transect estimates of abundance is less than that for strip transects
when animal speed is less than 200% of observer speed. This is due to the detection process. In
strip transects, animals entering the strip are detected at the boundary at which they enter
since detection probability is one inside the strip and zero outside. In line transects, the animals
that would be detected at the boundary in strip transects are unlikely to be detected as they ap-
proach from a distance with small detection probability. Furthermore, as detection probability
falls off gradually from the line some of these animals, if detected, may replace those that
moved further from the line that would, if immobile, have been recorded. So, the reduction in
the number of detections by the detection process reduces bias by a greater amount than the
bias in the estimated detection function augments it when animal speeds are slow. Conversely,
The Effect of Animal Movement on Line Transect Estimates of Abundance
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at animal speeds close to 200% of observer speed, bias in line transect sampling becomes simi-
lar to that in strip transect sampling due to the biased estimated detection function. At these
speeds, the detection function falls off very rapidly and many animals are assumed to be unde-
tected resulting in very high bias in abundance estimates which may exceed the bias in strip
transect sampling.
Bias is also related to the shape of the detection function (Fig. 5). At slow animal speeds,
bias is similar for all shape parameters considered (b= 2, 3, 5) which is unsurprising as the bias
arises from too many detections close to the observer and all three functions here have similar
shapes in this region (they all approximately equal one). At animal speeds exceeding observer
speed, the hazard rate with shape parameter b= 5 consistently produces smaller bias than the
other two shape parameters (b= 2, 3). This may be due to the assumed detection function
being isotropic: animals overtaking the observer are less likely to be detected ahead of the ob-
server when b= 5 than b= 2, 3 since the shoulder of the detection function is narrower. How-
ever, here we not only change the shape of the detection function but also the probability of
detection and so the bias may be a result of both effects.
2.2 Lookahead Distance. Fig. 6 shows that bias, at each speed, increases toward a plateau
as lookahead distance increases for a fixed detection function. The bias reaches a plateau since
the probability of detection at a large lookahead distance is small and so further increasing
Fig 3. Estimated detection function for line transect sampling for increasing animal speeds. Estimated
detection function averaged over 1000 simulations for line transect sampling (hazard-rate b= 2) with the
linear animal movement model at animal speeds 0% (black solid line), 20% (red dashed line), 40% (blue
dotted line), 80% (green dotdash line), 150%(purple longdash line) and 200% (grey twodash line)
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lookahead distance results in a smaller increase in detections—and so bias increases by less.
Also we see for small animal speeds, the bias increases slowly with lookahead distance, while
for animal speeds that exceed observer speed, bias increases rapidly. As for strip transect sam-
pling, this rapid increase is due to animals overtaking the observer from behind. For larger loo-
kahead distances, such animals remain in the encounter region for longer and are thus more
likely to be detected. Non-zero bias when the observer searches abeam (lookahead distance is
zero) is another consequence of animal speed exceeding observer speed, when animals travel
slower than the observer no bias occurs.
2.3 Truncation Distance. When analysing line transect data, it is common to truncate the
recorded distances at some truncation distance to prevent outliers unduly influencing the
abundance estimate [1] (pp. 103–108). We explored the effect of truncation distance on bias
with the linear animal movement model and b= 2 hazard-rate. We assumed that observers
were actively searching out to distances beyond the truncation distance, and recorded distances
of detected animals from the line when they were first detected. Thus, truncation was assumed
to be applied by the analyst, not in the field.
Results indicate little if any relationship between bias and truncation distance (Fig. 7). This
is unsurprising as bias is caused by too many detections near to the line which truncation does
not affect.
Fig 4. Frequency of detections at each perpendicular distance from the line. Estimated histogram of
recorded distances averaged over 1000 simulations for line transect sampling (hazard-rate b= 2) with linear
animal movement at animal speeds 0% (black solid), 20% (red dashed) 80%(green dotted) and 150% (purple
dotdash)
doi:10.1371/journal.pone.0121333.g004
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Discussion
Animal movement independent of the observer generates bias in abundance estimates from
line transect surveys. The degree of bias depends both on animal movement and dectectability.
For a given mean animal speed, the analytical strip transect expression provides an upper
bound for this bias across all possible detection functions and movement types. The expression
shows bias to be reduced when the observer searches further to the side and a shorter distance
ahead (smaller lookahead). Also, the result shows the significant bias caused when animals
travel faster than the observer [1] (pp. 131). To reduce bias in strip transect sampling, observers
should record only those animals that were initially inside the strip and ignore those animals
that enter from the side or behind [24]. As line transects are often much wider than strip tran-
sects and detection of animals is uncertain [25], it is not possible to exclude animals that enter
the strip from the side, but often animals that overtake the observer can (and should)
be excluded.
Bias in line transect sampling for most animal speeds is considerably less than that of strip
transect sampling. The magnitude of the bias depends on the behaviour of the surveyed animal:
those that quickly travel long distances cause more bias than those that travel slowly or remain
within a small area [16]. Once again, survey procedure can reduce bias in line transect surveys
by searching further to the side and a shorter distance ahead; however, this may increase the
Fig 5. Percentage bias in abundance estimate for different shape parameter values. Percentage bias in
abundance estimate for linear movement against animal speed as a percentage of observer speed for shape
parameters b= 2 (black solid), b= 3 (blue dashed) and b= 5 (red dotted) with percentage bias for a strip
transect (grey solid) estimated from 1000 simulations included for comparison.
doi:10.1371/journal.pone.0121333.g005
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chances of failing to spot an animal before it moves in response to the observer [26] (pp. 129–
131). If the animals to be surveyed tend to flee from the observer or be attracted, searching fur-
ther along the line will reduce the bias caused by this responsive movement [1] (pp. 32). Clear-
ly, this opposes our advice concerning independent animal movement. Often the bias caused
by independent animal movement will be small compared to that caused by movement in re-
sponse to the observer [27].
How to reduce bias from both independent and responsive movement will depend on the
type of survey and species. For aircraft surveys, animal speed is typically slow relative to observ-
er speed [28], and search effort is generally concentrated to the side rather than ahead. Thus in-
dependent animal movement is unlikely to cause bias. For shipboard surveys, observers tend to
concentrate more effort on searching ahead rather than to the side. This has the advantage that
animals are less likely to respond to the ship before detection, but greater bias will be generated
by independent animal movement. Therefore ship speed should be such that bias from inde-
pendent animal movement is low. For fast-moving animals, such as seabirds in flight, this may
not be an option; then methods for correcting for movement should be considered. For exam-
ple, seabirds on the sea might be surveyed, then a correction adopted based on the proportion
of time birds are in flight. If seabirds don’t respond to the ship, then one strategy to eliminate
bias is to simply record distances of detected birds when they pass abeam; if they do not pass
abeam while visible, then they are not recorded [1] (pp. 198–203). Corrections have also been
Fig 6. Percentage bias in abundance estimate against distance observer looks ahead. Percentage bias
against distance observer looks ahead with hazard rate b= 2 and animal speeds 50% (red dotted), 100%
(blue dashed) and 150% (black solid) of observer speed
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developed for seabirds in flight [15]. For terrestrial surveys of birds, surveying perched birds,
and separately estimating the proportion of time birds are in flight, can be effective [29]. For
surveys on foot of ground or arboreal animals, an assessment should be made of whether it is
feasible for observers to travel sufficiently fast without compromising other assumptions, to en-
sure bias from independent animal movement is low. If not, corrections for animal movement
should be considered.
Once the survey is completed there is, at present, no clear way to reduce bias in abundance
estimates using the standard methods. For example, truncating the data [1] (pp. 103–108) has
no discernible effect on the bias.
Here we have shown that bias can reach up to 10% for animal speeds of 50% –80% of ob-
server speed. This is useful for providing an idea of the severity of violating the assumptions of
distance sampling. However, the quantities of bias presented here are dependent on the type of
movement and survey protocol assumed. The expression for bias in strip transect sampling as-
sumes a linear movement of animals, which is unrealistic and causes greater bias compared to
other movement types [10]. Similarly, within the line transect simulation study we have not
considered how variable animal speed or heterogeneous home range radius may affect bias.
Furthermore, we only considered how the shape of the detection function affected bias by vary-
ing the shape parameter, which not only determines the function’s shape, but also the average
probability of detection; it may be more informative to describe the relationship between
Fig 7. Percentage bias in abundance estimate for different truncation distances. Percentage bias in
abundance estimate averaged over 1000 simulations for hazard-rate b= 2 against truncation distance
applied by the analyst for animal speeds 20% (red solid), 80% (green dashed) and 150% (purple dotdash)
doi:10.1371/journal.pone.0121333.g007
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detection function shape and bias for a fixed probability of detection. Also, in this study only
the hazard-rate detection function was fit to the simulated data, while in practice other detec-
tion functions may be chosen by model selection [1] (pp. 110), which may affect bias.
In conclusion, independent animal movement can cause substantial bias in abundance esti-
mates and its possible effects should be considered in line transect surveys. It arises from both
the detection process inherent in line transect sampling and from the movement characteristics
of the surveyed animal. There are, at present, no analytical methods to deal with this reality;
however, bias is reduced by searching further perpendicularly to the line, searching less ahead
and ignoring animals that overtake the observer.
Supporting Information
S1 Code. Computer Code for Simulation Study.
(ZIP)
Acknowledgments
The manuscript was peer-reviewed by two helpful and insightful anonymous reviewers.
Author Contributions
Conceived and designed the experiments: RG STB LT. Performed the experiments: RG. Ana-
lyzed the data: RG STB LT. Contributed reagents/materials/analysis tools: RG STB LT. Wrote
the manuscript: RG STB LT.
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