Incorporating animal movement into distance sampling

This paper is currently (27 September 2017) under peer-review at a journal. Please check with the authors for a later version before citing.

R. Glennie1 , S. T. Buckland1 , R. Langrock2 , T. Gerrodette3 , L. T. Ballance3 , S. J. Chivers3 , M. D.

Scott3 , and W. F. Perrin3
1
University of St Andrews, Centre for Research into Ecological and Environmental Modelling
2 Bielefeld University
3 Southwest Fisheries Science Center, NOAA Fisheries
4 Inter-American Tropical Tuna Commission, La Jolla, CA

Abstract triangular distribution. Given this, the decline in the num-

ber of detections as distance from the observer increases
Distance sampling is a popular statistical method to esti- is solely due to a change in the probability of detection;
mate the density of wild animal populations. Conventional thus, the recorded locations are used to estimate this prob-
distance sampling represents animals as fixed points in ability and, ultimately, animal density. In short, distance
space that are detected with an unknown probability that sampling is a thinned point process model with unknown
depends on the distance between the observer and the an- thinning probability (Hedley and Buckland, 2004; Yuan
imal. Animal movement, responsive or non-responsive to et al., 2016). The assumption that the survey is a tem-
the observer, can cause substantial bias in density estima- poral snapshot of the animal population is central to dis-
tion. Methods to correct for responsive animal movement tance sampling theory; the method, however, is applied to
exist, but none account for non-responsive movement in- surveys of mobile animal populations where transects are
dependent of the observer. Here, an explicit animal move- surveyed over a time interval within which animals may
ment model is incorporated into distance sampling, com- have moved a significant distance.
bining distance sampling survey data with independently
Animal movement can be in response to an observer’s
obtained animal telemetry data. A detection probability
presence (Turnock and Quinn, 1991), attraction or avoid-
that depends on the entire unobserved path the animal
ance, or can be of the animal’s own accord, independent
travels is derived in continuous space-time. The intractable
of the observer. Responsive movement is a well-known
integration over all possible animal paths is approximated
problem, and specific survey techniques, searching further
by a hidden Markov model. A simulation study shows the
along line transects to see animals before they respond
method to be negligibly biased (less than 5%) in scenarios
or remaining at point transects long enough for animals
where conventional distance sampling overestimates abun-
to resume normal behaviour, are recommended to mit-
dance by up to 100%. The method is applied to a line
igate bias in density estimates (Buckland et al., 2005).
transect survey of spotted dolphins (Stenella attenuata at-
Furthermore, double-observer methods exist that can ac-
tenuata) in the eastern tropical Pacific.
count for responsive movement (Palka and Hammond,
2001). In comparison, movement independent of the ob-
server has received little attention. For point transects, a
1 Introduction snapshot method is recommended to reduce bias in esti-
mates (Buckland, 2006); however, many surveys do not
Distance sampling is a statistical method used to esti- employ this method, and it does not suit technological ad-
mate the population density of wild animals (Buckland vances where observation technology may survey a point
et al., 2015). It is applied to a wide variety of taxa, e.g., for a considerable time. Splitting continuous surveying pe-
seabirds, cetaceans, primates, and ungulates. Many con- riods at points into discrete snapshots involves subjective
servation and management studies depend on the accuracy judgments that can affect the inferences obtained (Howe
of distance sampling inference; yet, the statistical method et al., 2017). Alternatively, cue-counting is used as it is not
relies on a key assumption that is significantly violated in biased by non-responsive animal movement, but this relies
many applications. on the animal population having a clearly defined cue. For
Distance sampling is a snapshot method: the survey is line transects, a rule-of-thumb, based on a limited simu-
assumed to occur instantaneously. Animals are idealised lation study, deems surveys on animals that move at less
as static points that are detected with unknown proba- than half the observer’s speed to be free of substantial
bility by an observer, who stands at a point or traverses bias (Hiby, 1982); yet, observer speed is often constrained
a line within the study region. Surveyed transects, lines by the transport chosen and the terrain covered. Thus,
or points, are placed according to a randomised design density can be unavoidably overestimated due to animal
such that animals are distributed independently of the movement. This overestimation is not caused by counting
observer. For line transects, animals are distributed uni- the same animal more than once, but by more animals
formly around the line, in point transects they follow a entering the transect from outside and recorded locations

1
leading to a biased estimated detection function. Surveys animal movement data is presented. The continuous-
of mobile animals record greater numbers of unique in- spacetime likelihood and its discrete approximation are
dividuals, compared to a hypothetically immobile popula- described. A simulation study compares this method to
tion, and animals are recorded closer to the observer; both conventional distance sampling, and the method is applied
effects led to positive bias in density estimation (Glen- to a line transect survey of spotted dolphins (Stenella at-
nie et al., 2015). This calls into question inference drawn tenuata attenuata) in the eastern tropical Pacific (ETP)
from surveys where animal movement is undeniable, and (Gerrodette and Forcada, 2005; Gerrodette et al., 2008).
precludes the use of distance sampling on populations of
fast-moving animals and on studies where transects are
surveyed over a long time period. 2 Methods
Previous work has considered only how movement affects
the number of animals seen, not where they are seen Suppose n animals are detected in total over the survey.
(Yapp, 1956). Random encounter models (Lucas et al., The goal is to estimate the total abundance in the sur-
2015), where animals are assumed to move in randomly- vey region, N . In distance sampling surveys, observers
orientated straight lines at constant speed (Hutchinson search transects and record the location of any animal
and Waser, 2007), can provide estimates of density, cor- they encounter. In line transect sampling, the perpendic-
rected for movement, given the count of animals seen ular distance from the line to each encounter is recorded;
and an independent estimate of animal speed. A detec- in point transects, the radial distance is recorded. Use of
tion probability can also be included, but must be ascer- two-dimensional location data, forward distance on line
tained independently. An advantage of distance sampling transects and angle of detection on points, is seldom used
is that the detection probability can be estimated from (Borchers and Cox, 2017). Furthermore, the time of a de-
the data. Yet, this probability, when movement is admit- tection, though routinely recorded, is not used. Here, a
ted, depends on the entire path the animal has travelled model is developed that uses both pieces of information:
whilst the transect is surveyed; this path, other than the for the ith detected animal, let ~xi be the two-dimensional
single location observed when the animal is detected, is location of the animal, when detected, relative to the ob-
unobserved. Thus, any estimation of detection probability server and ti be the time between the observer beginning
that accounts for animal movement must include a con- to survey the transect that animal i was seen on and the
tinuous space-time hidden process to describe the animal’s time animal i was detected.
trajectory. Furthermore, to calculate the proportion of an-
imals never seen, the method must average over all possi- Recording the single location an animal is encountered
ble animal trajectories. This can be achieved by specifying provides no information about how the animal moves: in-
an explicit model for the encounter process (Gurarie and dependent data is required on animal movement. Suppose
Ovaskainen, 2012). m animals are tracked or tagged and their movement paths
recorded over time. It is assumed the movement of these
Hidden Markov models (HMMs) (Zucchini and MacDon- tagged animals is representative of the movement of any
ald, 2009) are used for time series data that arise from animal in the study area. Note it is not required that the
an unobserved (or partially observed) stochastic spatial tagged animals be members of the surveyed population.
process. In particular, HMMs are used to analyse ani-
mal telemetry data (Langrock et al., 2012), animal loca-
tions recorded over time, where the paths taken by an- 2.1 Model
imals between recorded locations are averaged over ac-
cording to the movement model specified. Distance sam- In conventional distance sampling (CDS), the probability
pling observations consist of a single such recorded loca- density function (pdf ) of the recorded animal locations is
tion and the animal’s path until detection is unobserved; estimated. A detection function, g(x), is defined as the
thus, if independent information on animal movement is conditional probability an animal is detected given it re-
collected, distance sampling can be viewed as a HMM, sides at location x. For line transects, location is defined as
where animal paths are a hidden process and detection the perpendicular distance the animal is from the line; for
is the observed process. Tagging and tracking of animals point transects, it is defined as the radial distance from the
is becoming more common as the technology reduces in point. The probability density of the observed distances is
price and size. HMMs can allow this auxiliary information then given by
to improve distance sampling estimation. Pedersen et al.
(2011) developed a spatial HMM with an unobserved, dif- g(x)λ(x)
f (x) = R
fusive movement process which is described by a stochastic P
g(x)λ(x) dx
partial differential equation. The intractable continuous-
spacetime likelihood is approximated by discretising space where P is the set of all animal locations and λ is the
into a large number of spatial cells (Eydeland, 1994); the probability density function of the animal’s location. CDS
computations involved are costly, constraining the level of makes the design-based assumption that transects are
discretisation attainable. placed according to a randomised scheme. This implies
for line transects that λ is the density of a uniform distri-
Here, a spatial HMM that incorporates animal move- bution over P and for point transects a triangular distri-
ment into distance sampling using independently obtained bution over P.

2
A convenient functional form is chosen for the detection makes Λ, the probability of a given path occurring, math-
function and its parameters estimated by maximum like- ematically tractable: it is a Gaussian probability distri-
lihood. This form is chosen ad hoc to be half-normal or bution. For telemetry data, one can condition on the ini-
exponential with some polynomial adjustments. Alterna- tial location of the animal and compute the likelihood, LΛ
tively, an explicit model for the detection process can be easily (Okubo and Levin, 2013); for distance sampling sur-
specified by a two-dimensional hazard-rate function h(x, t) veys, the initial locations are assumed to be independently
(Borchers and Cox, 2017; Skaug and Schweder, 1999), distributed with respect to the transect.
which describes the detection rate of an animal residing
at point x at time t. This is equivalent to a survival pro- From the distance sampling survey, given animal i was
cess where death is interpreted as detection and the de- recorded in location ~xi at time τi , the likelihood for the
tection intensity varies over time and space. In practice, detection parameters, θ, is obtained by averaging over all
CDS analyses do not use the time of detections, a detection possible animal paths:
function can be derived from a given hazard by integrating
over the time interval that an animal is at risk of detec- n
R
tion. Nonetheless, here, the term CDS is used to refer to Y χi
g(~x, τi ) dΛ(~x)
Lθ =
2D hazard models also. The hazard commonly depends on i=1
pi
the radial distance between the observer and the animal,
r(~x, t), such that the hazard is infinite at zero radius and
decreases with increasing radius. where χi is the space of all measurable paths that
pass through location ~xi at time τi , and pi = 1 −
Here, the recorded detection times and the two- R S (~x) dΛ(~x) is the probability the encounter with an-
χ Ti
dimensional recorded location are used to estimate the imal i occurs at some time when the transect is surveyed
search process. Detection times are required since animals for total time T .
i
recorded at later times on the transect have had longer to
move and so may have originated at a further distance. Assuming the distance sampling survey and the animal
Rather than condition on the animal residing at a single telemetry data are independent, the combined likelihood
fixed point, the detection probability is derived conditional Lθ,λ = Lθ Lλ can be maximised to obtain maximum like-
on the animal travelling a fixed path over space. Given an lihood estimates, (θ̂, λ̂). Methods in section 2.2 describe
animal travels a path ~x and is seen at location ~xτ at time the approximations used to compute the likelihood. Max-
τ , the conditional pdf is imising the combined likelihood means that uncertainty
in the movement parameters is propagated to the den-
g(~x, τ ) = Sτ (~x)h(~xτ , τ )
sity estimation. Abundance can be estimated using either
where approach already available in distance sampling: a model-
 Z t
based estimator or a Horvitz-Thompson-like estimator.

St (~x) = exp − h(~xs , s) ds
0
The Horvitz-Thompson-like estimator of abundance is
is the probability of the animal eluding detection until
time t. Notice, detection probability now depends on time n
X 1
and the entire trajectory of the animal. For brevity, we N̂ =
p̂
i=1 i
term the model presented here MDS: movement in dis-
tance sampling.
Similar to CDS, if in place of assuming the distribution of A sandwich estimator for the variance of N̂ can be derived
animals, we assume the movement process is known, and analogously to the estimator used in CDS (Fewster et al.,
in place of conditioning on the location of an animal, we 2008).
condition on the path an animal has taken, the pdf of the
observed encounter on a transect of duration T is given Alternatively, an explicit model for abundance can be in-
by: tegrated (Buckland et al., 2016). For example, the Poisson
g(~x, τ )Λ(~x) process:
f (~x, τ ) = R R T (1)
χ 0
g(~y, t) dt dΛ(~y)
where Λ is the probability measure over all (measurable) (N penc )n exp(−N penc )
paths, χ, that an animal could have taken, that is, Λ(~x) is LN =
n!
the probability of path ~x. The denominator is the proba-
bility an animal is seen at some time on the transect and
where penc is the average probability of detection during
is required as we do not observe those animals that were
the survey. This gives a model-based estimate of abun-
never encountered.
dance by maximising the likelihood:
Yet, the paths of animals are unobserved and distance
sampling surveys provide no information on how animals
move. Thus, independent animal movement data is re- n Z
quired to determine Λ. Here, animal movement is de- N n exp(−N penc ) Y
Lθ,λ,N = Lλ g(~x, τi ) dP(~x)
scribed by diffusion with average speed parameter ν. This n! i=1 χi

3
2.2 Computation 2.3 Simulation study

The continuous space-time likelihood is analytically in- A simulation study is conducted to demonstrate the per-
tractable. The integral over all possible animal paths can formance of MDS compared to CDS for two particular
be approximated by quadrature. A buffer region around distance sampling surveys. The magnitude of the bias in
each transect is discretised into K cells of length and width CDS depends on the relationship between relative animal
∆x and time is discretised into steps of duration ∆t. An- speed, transect width, and the shape of the detection func-
imal movement occurs at these time-steps, thus animals tion (Glennie et al., 2015). This simulation study considers
are stationary within time-steps. the effect of animal speed when all other factors are fixed.

Notice that all integrals to be approximated are of the A study population of 100 animals in 100 square kilome-
form Z tres is simulated. Animals move according to a diffusion
process with average speed varying from 0.5 metres per
St (~x) dΛ(~x)
χ
second to 4.0 metres per second.

for some time t. Approximating animal movement by Two distance sampling surveys were simulated on this
jumps at each time-step, the function S can be separated population: a line transect study and a point transect
into components: study. The line transect survey consists of 50 transects
of length 1 kilometre; the observer traverses each line at
St (~x) = s0 (~x0 )s∆t (~x∆t )s2∆t (~x2∆t ) . . . st (~xt ) speed 1 metre per second. Hence, simulation scenarios
 R  cover relative animal speeds of 50% to 400% the speed
where sl (~x) = exp − l
l+∆t
h(~x, u) du is the probability of the observer. For the point transect survey, 100 points
were surveyed, each for 5 minutes.
of eluding detection during a single time-step given the
animal is at location ~x. Notice, observer movement can be In both surveys, the hazard of detecting an animal at a
integrated analytically within each time-step, thus only radial distance r is given by αr−β for parameters α, β > 0.
animal movement is discretised. Let Pt be a K × K diago- Detection parameters were chosen such that for a hypo-
nal matrix with k th diagonal st (~yk ) where ~yk is the centre
thetically immobile animal population, the effective area
location of spatial cell k. searched was approximately 0.015 square kilometres. This
corresponds to a line transect with half-width 30 metres
The integration over all paths is replaced with a sum over and point transect with radius 100 metres.
all paths on the discrete grid. When discretised, diffusive
animal movement can be approximated by a continuous- The independent animal telemetry data required was sim-
time Markov chain (Pedersen et al., 2011). The transition ulated from the diffusion process, recording the location
rate matrix of this chain, G, is block-Toeplitz in struc- of ten tagged animals every minute for one hour.
ture, and computing the matrix exponential, to obtain the
One hundred simulations were performed. A distance sam-
transition probability matrix Γ = exp(G∆t), is computa-
pling model with no animal movement was fit to each sim-
tionally demanding.
ulated data set. For fair comparison, a two-dimensional
The integral is approximated by a spatial HMM likelihood: hazard was used in this model also, unlike CDS mod-
Z els where a one-dimensional hazard is commonly used. A
MDS model, as presented in this paper, was fit to each
St (X) dΛ = p0 P0 ΓP1 Γ . . . Pt Γ1 (2)
χ data set with the auxiliary movement data.

where p0 is 1×K row vector representing the initial distri- The relative bias, mean square error and confidence in-
bution of animals on the grid with respect to the transect terval coverage was estimated for each scenario, with and
and 1 is a K × 1 column vector of ones. without movement incorporated.

In distance sampling surveys, the range of detection dis-

tances can be small compared to the distances an animal 2.4 Application: spotted dolphins
can travel during a survey period. The former makes it
necessary that ∆x be adequately small while the latter The presented method is applied to a 2006 shipboard line
requires the buffer around the transect, and so the grid, transect survey conducted in the eastern tropical Pacific
to be large. Ultimately, K is large. This makes the matrix on spotted dolphins, estimating the abundance within the
calculations in equation (2) computationally demanding. core area, as defined by Gerrodette and Forcada (2005).
The block-Toeplitz structure of G can be exploited to ac-
celerate computations using a 2D discrete Fourier trans- Dolphin schools are treated as the individual unit of de-
form (Lee, 1986) and the Krylov subspace approximation tection and group size estimated separately. This is a stan-
to the matrix exponential (see Appendix). dard approach in CDS; however, incorporating movement
makes the approach more questionable. In particular, we
For spatial HMMs, there is no analytical bound on the assume that schools do not fuse or break-up during the
error of this approximation. Practical advice is to reduce time the observer surveys each transect. Furthermore, we
the discretisation until the inference obtained no longer assume the movement model, informed by tags on single
significantly changes. individuals, describes the movement of a school as a whole.

4
The radial distance and angle to each detected spotted coverage was nominal within 1% for all parameters and
dolphin school was recorded. Only sightings in Beaufort across all simulation scenarios.
state 2 or less were retained. Once an encounter occurs,
observers cease to survey and approach the detected school 170 ●

in ‘closing mode’. The times when surveying ceased and 150

●
resumed are recorded. This affects the detection proba-

Percentage relative bias

130
bility. Short breaks in effort cannot be ignored, nor can 110
●

the transect be subdivided. The former leads to underes- 90

●
timation of detection probability and the latter leads to 70 ●
overestimation. The following hazard accounts for this in- 50 ●
terrupted effort: 30
●

−β 10 ●
●
h(~x, t) = αr(~x, t) e(t) ●
● ●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−10 ●
●
where r(~x, t) is the radial distance at time t between the −30
●
●
observer and location ~x, and e(t) = 1 when the observer −50 ●
● ● ●
is on effort at time t and zero otherwise. Thus, animal −70
movement during off-effort time is accounted for. 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4
Mean animal speed (m/s)
The location of the ship was recorded every ten minutes. It
is assumed the ship travels in a straight line at a constant Figure 1. Percentage relative bias in estimated mean
speed between these records. The movement model does detection probability (left pane) and abundance (right pane)
not account for movement caused by ocean current; it is for conventional distance sampling (dashed line) and distance
assumed that the animals and ship are drifting in the same sampling with movement incorporated (solid line) against
direction and at the same rate, thus this movement has no animal speed (metres per second) estimated from 100
effect on their relative positions. simulations of a line transect survey of 50 transects with
truncation width 30 metres and observer speed 1 metre per
Independent tag data is collected on nineteen spotted dol- second. Dotted lines mark 5% relative bias.
phins (Scott and Chivers, 2009) providing fixed locations
at approximately 15 minute intervals over 1–2 days.
School abundance is estimated using a Horvitz-Thompson- ●
170
like estimator:
n
Percentage relative RMSE

X 1 150
N̂ = ●
p̂
i=1 i 130 ●

where n is the number of detected schools and pi is the 110

probability school i is detected. The total number of spot-
90
ted dolphins is estimated as N̂ ŝ where ŝ is the estimated ●

mean school size. 70

● ● ●
●
●
Goodness of fit is evaluated by a chi-squared test. The 50
●
● ●

perpendicular distance from the transect line is subdivided 30 ●

●
into discrete bins; the observed and expected number of ●
●
●
10 ● ● ● ●
● ● ● ● ● ● ● ●
sightings in each bin is compared. ● ● ● ● ● ● ● ● ●

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4

Mean animal speed (m/s)
3 Results
Figure 2. Percentage relative root mean square error
(RMSE) in estimated mean detection probability (left pane)
3.1 Simulation study and abundance (right pane) for conventional distance
sampling (dashed line) and distance sampling with movement
Line transect simulation incorporated (solid line) against animal speed (metres per
second) estimated from 100 simulations of a line transect
CDS estimators of detection probability and abundance survey of 50 transects with truncation width 30 metres and
had bias > 10% when animal speed exceeded 1.5 metres observer speed 1 metre per second.
per second and CDS overestimated abundance by > 100%
for speed > 3.5 metres per second. In contrast, MDS led to
< 5% bias for all scenarios (Figure 1). Mean square error
Point transect simulation
for CDS estimators was dominated by their bias; MDS
showed constant MSE across all animal speeds (Figure 2).
CDS point transect sampling behaved similarly with bias
Confidence interval coverage across all parameters for CDS > 10% for animal speed > 2 metres per second and bias
was less than 40% for speeds over 1.0 metre per second reaching 90% for speed around 4 metres per second. In-
and fell to 0% for speeds over 2.0 metres per second. MDS corporating movement reduced bias to < 5% across all

5
scenarios and MSE varied negligibly. CDS 95% confidence 3.2 Application: spotted dolphins
interval coverage was poor (< 45%) for all parameters
when animal speed exceeded 2 metres per second, while
Estimated abundance of spotted dolphin schools in the
coverage was nominal for all parameters when movement
core area, as defined by Gerrodette and Forcada (2005),
was incorporated.
differed between models (Table 1). Mean group size was
164 with large variability due to the rare detection of very
large schools (> 2000 individuals). Incorporating move-
ment reduced the abundance estimate by 22% (33292 ani-
90 ●
mals). The coefficient of variation (CV) for the abundance
70
estimator was reduced (by 7.4%) when animal movement
Percentage relative bias

● was accounted for. The average speed of the ship was

50
●
17km/h; the estimated average speed of each spotted dol-
●
phin was 7.4km/h. The large reduction in the abundance
30
● estimate indicates that even though the dolphins move
●
10 ● relatively slowly compared to the ship, bias can be sub-
● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●
● ●
● ● ● ● ● stantial, because, whilst being surveyed, they can move
−10 ●
●
a large distance compared to the width of the transect.
●
−30 This highlights the danger of assessing whether movement
●
● is a problem based solely on relative animal speed; MDS
−50 ●
can account for the interdependent effects of animal speed,
●
transect width, and detection function shape.
0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4
Mean animal speed (m/s) For comparison, the expected number of sightings within
each 0.5 kilometre from the transect line was calculated
Figure 3. Percentage relative bias in estimated detection (Figure 5). MDS had a similar goodness of fit as CDS to
probability (left pane) and abundance (right pane) for the data (chi-squared test gives p-value of 0.31 for CDS
conventional distance sampling (dashed line) and distance and 0.35 for MDS); however, the estimated detection func-
sampling with movement incorporated (solid line) against tion differs considerably between the two methods (Figure
animal speed (metres per second) estimated from 100
6). The CDS estimated detection function has a narrower
simulations of a point transect survey with 100 transects of
shoulder and smaller detection scale indicating that ani-
radius 100 metres, surveyed each for 5 minutes. Dotted lines
mark 5% relative bias. mal movement has caused negative bias in the estimation
of detection probability. If the survey had indeed taken
place in a snapshot of time, CDS estimates the proba-
bility of an animal being detected, given it is inside the
transect, to be 0.47; MDS estimates this to be 0.62. Note,
this deficiency does not result in a marked difference in
goodness-of-fit to the observed data, but has an impor-
●
tant effect on the final abundance estimate.
Percentage relative RMSE

90
8

70 ●
●
● ●
6 ●
● ●
●
50 ●
●
● ● ●
Count

●
●
●
● 4 ●
●
30 ●
●
●
●
●
● ● ● ●
●
● ●
●
10 ● ●
● 2
● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ●

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4

Mean animal speed (m/s) 0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Figure 4. Percentage relative root mean square error Perpendicular distance (km)
(RMSE) in estimated mean detection probability (left pane)
and abundance (right pane) for conventional distance Figure 5. Observed number of spotted dolphin schools sighted
sampling (dashed line) and distance sampling with movement in each 0.5 km perpendicular distance from the transect line
incorporated (solid line) against mean animal speed (metres (shaded bars) with expected number of sightings from
per second) estimated from 100 simulations of a point transect conventional distance sampling model (dashed lines) and
survey with 100 transects of radius 100 metres, surveyed each distance sampling with movement (solid lines)
for 5 minutes.

6
 1.0
this information is often recorded in the field, but not used
in CDS models. Here, this information is essential and any
application of the model would require this data to be col-
Detection probability

lected. Furthermore, as for the ETP survey, periods of on

0.8 and off effort must be recorded. When the observer pauses
their efforts, animals continue to move; failing to record
this information leads to underestimation of the detection
probability. Records of the observer’s location over time
0.6
must also be kept. Furthermore, the relative location of
detected animals should be recorded relative to the ob-
server; this is contrary to practice in CDS where measure-
0.4 ments are made relative to the point or line. Also, animal
location must be recorded in two-dimensional space.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Perpendicular distance (km) Assumptions

Figure 6. Estimated detection function for a hypothetically The assumptions made in the theory presented are syn-
immobile spotted dolphin population for conventional distance onymous with those made in CDS. Violations of these as-
sampling (dashed line) and distance sampling with movement sumptions will cause bias in the inference obtained.
(solid line)
1. The path an animal travels is independent of the ob-
Estimate CV(%) LCL UCL server: animals do not respond to the observer and
CDS density 910 19.9 430 983 their movement is independent of the transect place-
MDS density 707 12.5 533 880 ment, that is, surveying does not preferentially take
place in areas animals would avoid or be attracted to.
Table 1. Maximum likelihood estimates of spotted dolphin
school density (per 106 km2 ) with coefficient of variation (CV) 2. Animals at zero radius are detected: this assumption
and lower and upper 95% confidence interval bounds for can be violated for animals that are not always avail-
conventional distance sampling (CDS) and distance sampling
able for detection, for example, a diving cetacean can
with movement (MDS)
be missed by an observer in a ship directly above it
(Borchers et al., 2013; Barlow, 2015).
4 Discussion 3. Location measurements are exact: this assumption
applies to observed locations of animals on the dis-
Distance sampling surveys on mobile animal populations tance sampling survey and the recorded locations of
should not ignore animal movement. The simulation study tracked animals. Observation error in animal teleme-
demonstrates the remarkable bias that non-responsive an- try data is common and can be accounted for (John-
imal movement can cause. Incorporating this movement son et al., 2008). Models for measurement error in
into distance sampling can mitigate this bias and remove distance sampling can also be incorporated (Marques,
the subjective judgment of when movement bias may have 2004).
occurred and to what extent.
4. Animal movement is diffusive: the simple model that
The cost of this improvement in estimation is the need for animal movement is a spatially-invariant, isotropic
additional information on animal movement. The expense diffusion process is violated by many animal popu-
and practicality of collecting such data depends on the lations; more realistic movement models can be con-
species to be surveyed. Tag data on cetaceans, ungulates, sidered. No matter what movement model is incorpo-
and seabirds is becoming more common. When no such rated, one assumes that all animals in the survey move
data is available, a ‘plug-in’ estimator of the movement according to the specified model. Departures from the
parameters could be used, similar to the multipliers used movement model could cause detection probability to
in CDS. Accounting for movement on imperfect knowl- be biased.
edge is better than ignoring it completely. The movement
5. Sampling is representative and independent: for the
information need not come from animals in the same pop-
distance sampling survey, this assumption requires
ulation as that surveyed by distance sampling, nor be col-
transects be placed according to a randomised de-
lected in the same time period. Nevertheless, it is assumed
sign, that transects be independent, and that animals
tagged animals behave as representative members of the
be independent. For animals that travel in groups,
study population; thus, it is recommended that movement
treating groups as the independent unit to be sampled
information be collected around the same time the dis-
may be a better choice. For the movement model, it
tance sampling survey is conducted and on animals that
is assumed that tagged animals move independently
are members of the study population.
and that the sample of tagged animals be representa-
Estimating a detection process that depends on time nec- tive of the surveyed population. One can use teleme-
essarily requires detection times be recorded. In practice, try from tagged animals who are not members of the

7
 surveyed population, but only with the assumption Finally, the methods presented can be extended to double-
these animals exhibit movement patterns similar to observer distance sampling, where two observers survey
those animals surveyed by distance sampling. the transect simultaneously. If multiple sightings of an in-
dividual by different observers can be matched together,
then only animal paths that pass through these multiple
Model extensions observed locations need be considered, providing informa-
tion on animal movement directly from the distance sam-
The model formulation is flexible and can include exist- pling data and improving the estimation of each animal’s
ing extensions of conventional distance sampling. Here, detection probability.
only a hazard that depends on radial distance was consid-
ered. A hazard that depends on angle and radius could be
used. For point transects, any detection function can be Conclusion
used to define a hazard (Borchers et al., 2015). Yet, for
line transects, the method relies on an analytical form for Animal movement can be incorporated with distance sam-
St : observer movement makes integration of the hazard pling. The presented theoretical framework provides a ba-
more complicated. This could be avoided by discretising sis for further development and the computational ap-
observer movement similarly to animal movement, that is, proach discussed makes the method applicable. Account-
the observer only moves at time-steps and not within time- ing for animal movement can mitigate the bias it causes
steps; however, this can introduce significant bias when in the surveys where distance sampling is applied, and it
discretisation is rough and often requires much lower lev- can widen the application of distance sampling to animals
els of discretisation than otherwise necessary. whose movement has so far prohibited its use.
Another popular extension is multiple-covariate distance
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