Appl. Statist. (2011)
60, Part 3, pp.
Estimating the density of Antarctic krill (Euphausia
superba) from multi-beam echo-sounder
observations using distance sampling methods
Martin J. Cox and David L. Borchers,
University of St Andrews, UK
David A. Demer and George R. Cutter
Southwest Fisheries Science Center, La Jolla, USA
and Andrew S. Brierley
University of St Andrews, UK
[Received November 2009. Revised July 2010]
Summary. Antarctic krill is a key species in the Antarctic food web, an important prey item
for marine predators and a commercial fishery resource. Although single-beam echo-sounders
are commonly used to survey the species, multi-beam echo-sounders may be more efficient
because they sample a larger volume of water. However, multi-beam echo-sounders may miss
animals because they involve lower energy densities. We adapt distance sampling theory to
deal with this and to estimate krill density and biomass from a multi-beam echo-sounder survey.
The method provides a general means for estimating density and biomass from multi-beam
echo-sounder data.
Keywords: Acoustic; Antarctic krill; Distance sampling; Multi-beam echo-sounder
1.
Introduction
Antarctic krill (Euphausia superba; hereafter simply called krill) are a key component in the
Antarctic food web (Atkinson et al., 2001). The abundance and swarming behaviour of krill
make it important to many marine predator species and as a commercial fishery resource;
swarms can occur with thousands of individuals per cubic metre of sea and contain thousands
of kilograms of krill. Management of the competing interests of marine predators and fisheries
requires regular monitoring of krill abundance. Ship-based remote sensing acoustic surveys,
which are conducted by using single-beam echo-sounders (SBEs), are key to this, with surveys
frequently covering large areas (Brierley et al., 1999).
Echo-sounders function by transmitting a sound pulse with known characteristics via a transducer into the column of water and recording the characteristics of the sound, or echo, that is
scattered back towards the transducer. The intensity of the backscattered acoustic echo can be
scaled by the use of appropriate acoustic target strength models (e.g. Demer and Conti (2005)
for krill) to estimate the density of animals in the volume of water that is sampled by the echosounder. Modern scientific SBEs operating at acoustic frequencies of 120 and 200 kHz typically
Address for correspondence: Martin J. Cox, Pelagic Ecology Research Group, Gatty Marine Laboratory, University of St Andrews, St Andrews, KY16 8LB, UK.
E-mail: [email protected]
© 2011 Royal Statistical Society
0035–9254/11/60000
2
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
have a narrow beam width of 7◦ . In contrast, multi-beam echo-sounders (MBEs) have much
wider swath widths (in this case 120◦ ) that are often made up of more than 100 narrow beams
(individual beam widths in this case of 1:5◦ × 2◦ ) and so sample a much larger volume of water.
This is particularly useful when studying organisms with aggregative behaviour, such as krill,
since entire swarms can be acoustically sampled (see Gerlotto et al. (1999)).
Methods that are used to estimate krill density from SBE data typically assume that all
swarms within the SBE beam are detected (e.g. Brierley et al. (1999) and Reiss et al. (2008),
but see Demer et al. (1999)). This may be a reasonable assumption in the case of SBEs because
of the concentration of acoustic energy in a narrow beam. It is a less tenable assumption in
the case of MBEs, where the acoustic energy density in the wider swath width is substantially
lower. Swarms at greater distances from the transducer may be missed and, in this case, a model
for the probability of detecting a swarm as a function of distance is required for unbiased density estimation. In this paper we extend and apply distance sampling methods to estimate the
probability of detection and density of krill from MBE data. An extension of standard distance
sampling theory is required because the assumption of standard theory that targets (swarms)
are uniformly distributed fails in the acoustic survey context.
In Section 2, we describe the key data obtained from the MBE survey. We summarize standard distance sampling theory and consider the validity of its key assumptions in the context
of acoustic surveys in Section 3. Section 4 describes extensions to the theory to deal with the
failure of a key assumption of standard theory on acoustic surveys and develops the associated
likelihood function and maximum likelihood estimator. The bias of the maximum likelihood
estimator is investigated by simulation in Section 5. The method is applied to estimate krill
density and biomass by using data from an MBE survey off the South Shetland Islands in
Section 6.
2.
Data
Krill swarms were sampled by using an SM20 MBE (acoustic frequency 200 kHz; Kongsberg
Mesotech Ltd, Vancouver, Canada). MBE data were collected along 41 line transects, lengths
varying from 2.5 to 3.5 km, from February 2nd–9th, 2006, near Cape Shirreff, Livingston Island,
South Shetland Islands, Antarctica (Fig. 1). Krill swarm boundaries were identified from the
data by using the acoustic processing software (Echoview v3.5, Myriax, Hobart, Australia).
Key data that are used in the analysis are shown in Table 1 and Fig. 2. The MBE mean volume
backscattering strength Sv (see MacLennan et al. (2002) for a formal definition) observations
were calibrated by using the technique that is described by Cox et al. (2010) and scaled by using
the estimate of krill target strength from the krill sound scattering model of Demer and Conti
(2005), from which swarm biomass b was estimated. (Sv is the sum of the acoustic backscattering
area (square metres) of organisms, in this case krill, per unit volume (cubic metres), in the logarithm domain, dB re 1 m−1 .) In common with many acoustic survey analyses (e.g. Brierley et al.
(1999) and Reiss et al. (2008)), we treat individual swarm biomass as being estimated without
error. A total of n = 1006 krill swarms were detected on the survey.
3.
Standard distance sampling
Distance sampling surveys are used to estimate abundance N or density D of objects (which
are called ‘animals’ henceforth) when some animals in the region searched are missed and
the probability of detection depends on the distance from the observer. The key assumptions
Estimating Density of Antarctic Krill
3
Fig. 1. Cape Shirreff study site: line transects are given in grey scale; C, detected krill swarm geometric
centres
Table 1. Description of swarm location data used in
this analysis†
Measurement
Swarm centre horizontal distance
Swarm centre depth
Swarm centre radial distance
Swarm centre angle
Truncation distance
Swarm biomass
Symbol
Units
x
y
r
θ
w
b
m
m
m
rad
m
kg
†Swarm centre location is measured relative to the MBE.
of standard distance sampling methods are as follows (from Buckland et al. (2001), pages
18–19).
(a) Survey lines or points are randomly placed with respect to animals, or systematically
placed with a random start point.
(b) All animals at distance 0 are detected.
(c) Animals are detected at their initial location.
(d) Measurements are exact.
The two main types of distance sampling survey are line transect and point transect. In the
case of line transect surveys, observers move along each of a set of lines and survey a twodimensional region within some distance w of the lines; in the case of point transect surveys,
observers survey a fraction of a two-dimensional circle about each of a set of points. Although
4
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
Fig. 2. MBE swath geometry and location measurements .ri , θi / for the i th krill swarm detected (one half
of the swath is shown; larger acoustic volume backscattering strength values are shown as darker samples;
data at radial distances beyond w are truncated; y Å is the depth at which the width of the swarm is a maximum
in the x -dimension; y D W sin.π=3/; , rectangle for which the expected probability of detecting a krill swarm
is estimated (see equation (3), Section 4)): 1, swath vertical axis; 2, krill swarm with the boundary of the
swarm delineated by using acoustic processing software (, geometric centre of the swarm); 3, seabed; 4,
sea surface (x -axis); 5, maximum swath observation angle .θ D π=3/
echo-sounder surveys involve an ‘observer’ (the echo-sounder) traversing lines, for distance
sampling analysis these surveys are more conveniently thought of as point transect surveys
sampling in depth and perpendicular-to-transect dimensions (compressed along the length of
the transect). The MBE samples a .2π=3/-rad segment of a circle about a point spanning the
length of the line (see Fig. 2).
Although assumptions (b)–(d) are defensible for MBE surveys, krill are known to exhibit
a non-uniform vertical distribution (Demer and Hewitt, 1995; Demer, 2004) and so assumption (a) is probably flawed. This is true even when vessel transects are located randomly on the
surface—because the transects are not located randomly with respect to depth.
Assumption (a) enables the distribution of animals in two-dimensional space to be treated as
known and uniform. Without this assumption the probability of detection g.r/ and distribution
of animals with respect to distance from the observer πr .r/ are confounded. To see this, consider
the point transect likelihood function for detected animals. Given radial distances r1 , . . . , rn to
the n detected animals, the likelihood for φ, the unknown parameter vector of g.r/, is (from
Buckland et al. (2001))
L.φ/ =
n
i=1
g.ri / πr .ri /
w
0
,
.1/
g.r/ πr .r/ dr
where w is the maximum search distance (which in our case is the seabed depth—see Fig. 2).
Because g.r/ and πr .r/ occur only as a product in the likelihood, they are not estimable separately. Hence standard distance sampling methods require πr .r/ to be known to estimate φ and
Estimating Density of Antarctic Krill
5
g.r/. In the next section, we describe an extension of standard methods which allows separate
estimation of πr .r/ and g.r/.
Given πr .r/ and an estimator ĝ.r/, the standard point transect estimator of animal density
from a survey in which a fraction γ of the circle is searched is D̂2 = n=as P̂ a , where
w
P̂ a = γ
ĝ.r/ πr .r/ dr
0
is the expected probability of detecting an animal given that it is within a distance w of the
observer (see below), and as = 2πw2 . In our context, D̂2 is an estimate of total density throughout the column of water in the along-surface (x) and depth (y) dimensions.
4.
Distance sampling for multi-beam echo-sounder surveys
As noted above, krill do not distribute themselves uniformly with respect to depth. To develop
an estimator which allows for this and the possibility that the probability of detection is a function of angle θ, we write the detection function as p.r, θ/ and consider the joint probability
density function (PDF) of radial distance r and angle θ of krill swarms, which can be written
as πr,θ .r, θ/ = πr .r/ πθ|r .θ|r/. Note that the πs here are function names, not the mathematical
constant. In common with standard distance sampling methods, we ‘fold’ the data about the
vertical (θ = 0) for detection function estimation. In this case, the expected probability of detecting an animal within distance w of the MBE which searches out to an angle of γπ rad either
side of the vertical can be written as
γπ w
Pa =
p.r, θ/ πr,θ .r, θ/ dr dθ:
.2/
0
0
In the conventional distance sampling scenario p.r, θ/ =g.r/ and πθ|r .θ|r/ = 1=π (rememberw
ing that we have folded about the vertical), so that Pa = γ 0 g.r/ πr .r/ dr, as given above, with
1
γ = 3 (see Fig. 2).
In our case, because we want to model a non-uniform krill depth (y) distribution, it is convenient to work in terms of .x, y/ rather than .r, θ/. By locating transects randomly on the sea
surface, we can treat the PDF of the horizontal distances from the line, given that they are
within w sin.π=3/ of the line, as known and equal to 1=w sin.π=3/ (see Fewster et al. (2008) and
Buckland et al. (2001), pages 232–235). We denote this PDF πx .x/. Random transect placement
also means that we can treat y as independent of x and write πx,y .x, y/ = πy .y/=w sin.π=3/. Here
πy .y/ is some smooth PDF with unknown parameter vector ϕ (see details in Section 4.4).
It is now convenient to consider the estimation of density within the rectangle that is defined
by 0 x w sin.π=3/ and 0 y w (see the broken line in Fig. 2) rather than the ‘pie slice’
0 r w and 0 θ γπ. To deal with the fact that no detections are made at radial distances
r > w and angles θ > γπ, we define p.r, θ/ to be 0 for r > w and for θ > γπ. We can then write the
expected probability of detecting a krill swarm in the rectangular area as
P=
0
w w sin.π=3/
0
√
p{ .x2 + y2 /, tan−1 .x=y/} πy .y/
1
dx dy:
w sin.π=3/
.3/
Given models for p.r, θ/ with unknown parameter vector φ and for πy .y/ with unknown
parameter vector ϕ, and that n krill swarms with centres .x1 , y1 /. . . , .xn , yn / were observed, the
likelihood for φ and ϕ is
6
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
L.φ, ϕ/ =
√
n p{ .x2 + y2 /, tan−1 .x =y /} π .y /={w sin.π=3/}
i i
y i
i
i
:
P
i=1
.4/
Although the above formulation is slightly different from that of Marques et al. (2010), the
central ideas are taken from that work.
4.1. Random seabed depth
The seabed depth varied along the transect line. Observations of seabed depth were available
every 100 m, generating 1065 depth observations within the survey region. We used these observations to model the probability that the sea at a randomly chosen location was no less than
y deep. We refer to this logistic function a.y, w/ as the depth attenuation function, and we use
this function to account for the non-uniform seabed (Fig. 1) throughout the survey region. It
models the proportion of the total transect length on which the depth was at least y. We model
it as
[1 + exp{−.y − β0 /=β1 }]−1
a.y, w/ =
.5/
T.w/
where
T.w/ =
w
[1 + exp{−.y − β0 /=β1 }]−1 dw,
0
and β0 and β1 are parameters to be estimated from the seabed depth observations. This functional form for a.y, w/ was chosen because it is bounded between 0 and 1, as is required for a
cumulative distribution function, and it proved to be adequate.
We found that models for πy .y/ with w equal to the maximum seabed depth provide poor fits
to the data. Because w in the model corresponds to a depth below which no krill can occur (rather
than bottom depth per se), we also considered models in which we estimated w. In these models,
we constrained w to be between the maximum observed krill swarm depth and maximum seabed
depth. Inclusion of a.y, w/ leads to the following likelihood in place of equation (4):
√
n p{ .x2 + y2 /, tan−1 .x =y /} π .y /.1=x
i i
y i
max / a.yi , w/
i
i
L.φ, ϕ, w/ =
.6/
Å
P
i=1
where
PÅ =
0
w xmax
0
√
p{ .x2 + y2 /, tan−1 .x=y/} πy .y/.1=xmax / a.y, w/ dx dy,
and xmax is the maximum rectangle width in Fig. 2, which is equal to w sin.π=3/.
We maximize the likelihood equation (6) to obtain maximum likelihood estimates of φ, ϕ
and w (taking β0 and β1 as known).
4.2. Krill density, abundance and biomass estimators
We estimate the number of krill swarms per unit volume of sea as
D̂s3 =
n
1
L 2xmax ŵP̂ Å
Å
where P̂ is P Å evaluated at the maximum likelihood estimators φ̂, ϕ̂ and ŵ.
.7/
Estimating Density of Antarctic Krill
7
Krill swarm areal density (the number of swarms per unit area of sea surface) is estimated by
D̂s = D̂s3 ŵ, and the krill swarm abundance in the survey region (with surface area A) is estimated
by N̂ s = AD̂s .
In standard distance sampling, estimates of individual density and abundance are usually
obtained by multiplying estimates of group density or abundance by estimated mean size of
group. Because there may be size selectivity in observing groups, the mean size of group is
conventionally estimated by regressing log(group size) against distance (or ĝ.x/), and using the
value of the regression line at distance 0 (where all groups are seen, by assumption). See Buckland et al. (2001), pages 73–75, for details of the method. We use a similar method to estimate
the mean biomass of the swarm and total biomass in the survey region.
We expect the mean biomass of observed swarms (the mean of b) to decrease at larger angles θ
because the probability of detection decreases with θ (see Section 6) and because parts of swarms
at larger θ may be undetectable. We therefore regressed the logarithm of observed swarm biomass
log.b/ against θ and used the estimated intercept α̂ and residual variance σ̂ 2 from the regression
to estimate the mean biomass of the swarm and to correct for bias (Ê[b] = exp.α̂ + σ̂ 2 =2/, where
b is the biomass of the swarm). A regression of log.b/ against radial distance r was found to
have zero slope.
The total krill biomass in the survey region was estimated by B̂ = N̂ s Ê[b] and the biomass
density was estimated by ρ̂ = B̂=A.
Variance and 95% confidence intervals were estimated by the non-parametric bootstrap with
1000 replicates, using a transect as the sampling unit. Confidence intervals were calculated
by using the percentile method. The survey contained 41 transects of varying length and the
transect selection probability in the bootstrap was made proportional to transect length.
To use the above estimators, we need suitable models for the detection function p.r, θ/, and
the krill depth distribution PDF πy .y/. We develop these in the next two subsections.
4.3. Krill detection function models
The probability of detecting a swarm is modelled as p.r, θ/ = g.r/ q.θ/. Standard distance sampling detection functions can be used for g.r/; we use the half-normal form, with a single
parameter φ1 :
2
−r
g.r/ = exp
:
.8/
2φ21
Without independent data on q.θ/, it and πy .y/ are confounded. Generally, MBE systems are
less likely to detect an acoustic target in the outer beams (those at larger θ) than in the centre
beams. The beam-by-beam sensitivity was estimated by using observations of MBE beam-bybeam noise. Noise was assumed to be isotropic and estimated by using Sv -data that were collected
at the Cape Shirreff study site.
The beam-by-beam logarithmic Sv (see for a definition MacLennan et al. (2002)) observations
with the MBE operating in ‘passive mode’ and transformed to the linear domain (Sv = 10Sv =10 )
and inverted so that beams with lower noise are more likely to detect krill swarms, giving a
relative measure of krill swarm detection probability. The passive mode Sv -data were collected
within the survey region, but not during the krill survey. The following hazard rate functional
form with parameters α0 , αθ and α1 was found to be adequate:
−α1 θ
:
.9/
q.θ/ = α0 1 − exp −
αθ
8
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
4.4. Krill depth distribution model
The density of krill swarms is known to be very low at the surface (Brierley et al., 1999) and is
expected to be low beyond 100 m (Demer, 2004):
(a) a left- and right-truncated normal distribution, y|w ∼ N.ϕ1 , ϕ22 /={F.w/ − F.0/}, where
F.w/ is the cumulative distribution function of y evaluated at w and F.0/ is the cumulative distribution function evaluated at 0;
(b) a right-truncated beta distribution,
πy
ϕ2 −1
y
y
Γ.ϕ1 + ϕ2 / y ϕ1 −1
1−
;
w =
w
Γ.ϕ1 / Γ.ϕ2 / w
w
(c) a right-truncated log-normal distribution, log.y/|w ∼ Nw .ϕ1 , ϕ22 /={F.w/ − F.0/}, where
F.w/ is the cumulative distribution function of log.y/ evaluated at w and F.0/ is the
cumulative distribution function evaluated at 0;
(d) a uniform distribution: πy .y|w/ = 1=w.
5.
Simulation testing
We conducted simulations to investigate the bias of the maximum likelihood estimator of abundance, simulating from a uniform πx .x/, the normal model for πy .y/ and the half-normal model
for g.r/ that were fitted to the survey data (see Section 6). Simulations were conducted for a
range of krill swarm abundances in the pie slice of Fig. 2 (Nsim ), ranging from Nsim = 100 to
Nsim = 2200, in increments of 50. For each Nsim 1000 simulations were performed.
The simulated sampling distribution of N̂ sim and its mean N̂¯ sim is plotted against Nsim in
Fig. 3(a). The relative bias (.Nsim − N̂ sim /=Nsim , where N̂ sim is the estimate for a given Nsim )
is shown in Fig. 3(b). Fig. 3 indicates that the abundance estimator is approximately unbiased
for Nsim greater than 500, which corresponds to an expected sample size of about n = 250. At
smaller sample sizes it is slightly positively biased. Fig. 3(a) also suggests that the estimator
has a substantially more skewed sampling distribution than a normal random variable, except
perhaps for large Nsim (2200 or more), corresponding to a sample size n of more than about
1200.
6.
Results
6.1. Seabed depth distribution model
The distribution of observed depths on the survey is shown in Fig. 4(a). The complement of
the empirical distribution function of depths .1 − CDF.y/, the observed proportion of depths
greater than y), together with the estimated probability that a depth is greater than y .a.y, w/;
see equation (5)) is shown in Fig. 4(b). The model provides a very good fit.
6.2. Angular detection function model
Data collected during the SM20 calibration exercise that was conducted at Cape Shirreff showed
an increased background noise on beams towards the edge of the MBE swath, as shown in
Fig. 5(a). The model for q.θ/ that is given in equation (9) provided a good fit to the inverse of
background noise (Fig. 5(b)) and showed very little loss in detectability at angles that are less
than θ = 0:7 rad, after which detectability dropped off rapidly.
9
1500
1000
0
500
^
Estimated abundance (Nsim)
2000
2500
Estimating Density of Antarctic Krill
0
500
1000
1500
2000
2500
0.4
Simulated abundance (Nsim)
(a)
●
●
●
●
0.0
0.2
● ●
–0.2
^
Relative bias (Nsim−Nsim)/Nsim
●
●
–0.4
●
200
400
600
800
1000
1250
1500
1750
2000
Nsim
(b)
Fig. 3. Simulated sampling distribution of N̂ s over a range of known swarm abundances (Nsim D 200–2200
in increments of 50; there were 1000 simulations for each Nsim ): (a) features of the simulated sampling
distribution of N̂ sim ( ) plotted against Nsim , with the mean N̂¯ sim ./ and ˙2 standard errors (5, lower bound;
4, upper bound) for each
p Nsim -value; (b) boxplots of the relative bias, .N̂ sim Nsim /=Nsim (the pair of curves
are at 1:96 s
e.N̂ sim /= 1000 above and below the mean N̂ sim )
1.0
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
0.6
0.4
200
0
0.0
0.2
100
Frequency
300
0.8
400
10
40
60
80
y,
(a)
100
120
0
50
100
150
(b)
Fig. 4. Estimated distribution of seabed depths, a.y/: (a) distribution of mean seabed depths over each
100-m segment of the transect line; (b) proportion of depths greater than y (- - - - - - - ) and the estimated a.y/
(see equation (5)) fitted to the data, β̂ 0 D 0:115 (coefficient of variation 18%) and β̂ 1 D 5:920 (coefficient
of variation 0.03%)
6.3. Vertical distribution and detection function models
Parameters of the krill vertical distribution PDF πy .y/ and radial distance detection function
g.r/ were estimated by maximizing likelihood equation (6) with respect to φ, ϕ and w, treating
â.y, w/ and q̂.θ/ as known. The krill vertical distribution model was selected by using Akaike’s
information criterion AIC and the goodness of fit was assessed by using a χ2 -test with the depth
intervals that are shown in Fig. 6. Expected values in each depth interval were obtained by
integrating under the full curve in Fig. 6. The truncated normal model for πy .y/ was selected on
the basis of AIC-weight (0.984) and was the only model with an acceptable goodness of fit. See
Table 2 and Fig. 6. For all πy .y/, the likelihood that maximized φ, ϕ and w consistently outperformed the likelihood that maximized only φ and ϕ (equation (4)), which had both higher
AIC-values and goodness-of-fit, χ2 , that were significant for all πy .y/.
6.4. Estimated krill density and biomass
Density, abundance and biomass estimates are shown in Table 3, together with the encounter rate and πy .y/ and g.r/ parameter estimates. The regression from which the mean swarm
biomass (E[b]) was estimated is shown in Fig. 7.
7.
Discussion
Investigations of the spatial distribution of krill at the South Shetland Islands during the austral
summer have shown that krill undertake diel vertical migration (DVM) (see Demer and Hewitt
(1995) and Demer (2004)) and reside mostly in the upper 40 m, making it extremely unlikely that
krill swarms have a uniform vertical PDF. At larger temporal scales to the west of the Antarctic
Peninsula, seasonal variation in krill biomass has been shown, with krill biomass located in
11
26
28
θ
20
22
24
Sv
m−1)
Estimating Density of Antarctic Krill
0
20
40
60
80
100
120
1.2
0.2
0.4
0.6
0.8
1.0
θ)
1.4
(a)
0.0
0.2
0.4
0.6
θ
0.8
1.0
(b)
Fig. 5. Variation in beam-by-beam sensitivity of the SM20 MBE that was used during the krill survey (the
model is given in equation (9); α̂0 D 1:12 (coefficient of variation 1.6%); α̂0 D 0:87 (coefficient of variation 1%);
α̂1 D 10:37 (coefficient of variation 14.1%)): (a) background noise, which increases (higher Sv ) towards the
edges of the swath; (b) inverse of the data in (a) (), folded about θ D 0, and the fitted angular detection
function q̂.θ/ (
)
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
0
0
50
50
100
100
150
150
200
200
250
250
300
300
12
0
20
40
60
80
20
40
80
100
60
80
100
0
0
50
50
100
100
150
150
200
200
250
250
60
(b)
300
(a)
300
0
100
0
20
40
60
(c)
80
100
0
20
40
(d)
Fig. 6. Model fits for various krill vertical distribution models (- - - - - - - ) (πy .y/, with parameter estimates ϕ̂1
and ϕ̂2 ) and half-normal radial distance detection function model (. . . . . . . ) (g.r/, with parameter φ1 ) (Ŵ D 100
m for all models;
, expected number of detections, conditional on the total sample size; πy .y/ and g.r/
have been scaled to correspond to the scale of the expected number of detections): the histograms show the
observed frequencies of the swarm in 10-m depth bins and the vertical distribution models are (a) normal
(ϕ̂1 D 61:1 m; ϕ̂2 D 27:8; ϕ1 D 76:6 m), (b) beta (ϕ̂1 D 2:27; ϕ̂2 D 1:70; ϕ1 D 78:6 m), (c) log-normal (ϕ̂1 D 5:23;
ϕ̂2 D 0:99; ϕ1 D 71:4 m) and (d) uniform (ϕ̂1 D 99:8 m) (the normal distribution was selected on the basis of
AIC
deeper water in the winter, and shallower in the summer (Lascara et al., 1999). Non-uniform
krill vertical distributions have been found at other locations also, e.g. around South Georgia
(54◦ S 35◦ W), where krill have been shown to undertake DVM that varies with location relative
to the continental shelf (Cresswell et al., 2007). It would be unrealistic to assume a uniform krill
swarm vertical distribution in general, but given the high variation in krill vertical distribution
it would also be unrealistic to prespecify a depth distribution model.
Estimating the krill vertical distribution is particularly important for estimating the accessibility of krill to air breathing predators: many krill predators are constrained in their depths
of diving, so being able to stratify the krill density estimates by depth will facilitate estimation
of biologically plausible prey fields. Furthermore, estimating krill swarm density will enable
Estimating Density of Antarctic Krill
Table 2.
13
Parameter estimates†
Krill vertical distribution
πy (y)
ϕ̂1
g(r), πy (y),
ŵ
φ̂1
AIC
ΔAIC AIC-weight
χ2
p-value
ϕ̂2
Normal
61.1 27.8
76.6
Beta
2.27 1.70 78.9
Log-normal 5.23 0.99 71.1
Uniform
—
— 126.4
100
100
100
100
7899.4
7907.7
7935.8
7996.1
0
8.3
36.4
96.7
0.984
0.016
0
0
0.2
0.02
0
0
†ΔAIC is the difference in AIC from the model with smallest AIC. The χ2 p-value
is for a χ2 goodness-of-fit test.
Table 3.
region†
Model parameter and krill biomass variance estimates for the survey
Parameter
Normal ϕ̂1
Normal ϕ̂2
Half-normal φ̂1
ŵ
Encounter rate n=L (km−1 )
Å
P̂
N̂ s
Ê[b] (kg)
B̂ (tonnes)
ρ̂ (g m−2 )
Point
estimate
Coefficient-ofvariation
estimate
95%
confidence
interval
61.10
27.76
76.62
100
9.37
0.45
1597
211.54
227.86
22.37
0.13
0.11
0.77
0.01
0.17
0.13
0.28
0.17
0.35
0.35
(47.93, 75.54)
(21.91, 32.38)
(66.3, 500)
(98.6,100)
(5.97, 11.75)
(0.20, 0.63)
(1244, 3220)
(149.57, 290.12)
(224.71, 768.23)
(15.34, 52.43)
†Coefficients of variation were calculated by the non-parametric bootstrap (1000
replicates) with the transect as the sampling unit; confidence intervals were calculated by using the percentile method.
researchers to examine the energetic value of krill swarms, which is important for estimating
krill predator foraging strategies; this can be done by examining the energetic interconnectivity
of swarms by applying percolation theory, for example (e.g. Reynolds et al. (2009)).
Given that ambient levels of light are believed to trigger DVM (e.g. Zhou and Dorland (2004)),
it may be possible to parameterize the vertical distribution function πy .y/ to include ambient
levels of light at the surface as a parameter. However, this approach would require the likelihood
to be rewritten and recent research suggests that DVM may not be a ubiquitous feature of krill
behaviour. Tarling et al. (2009) found that ambient levels of light did not influence swarm depth
in open ocean. Tarling et al. (2009) suggested that distance to land-based air breathing krill
predator colonies may explain the magnitude of DVM, hypothesizing that krill outside predators’ foraging range are not required to seek refuge in deeper waters and so do not undertake
DVM. There are many potential additional parameters for both the vertical distribution PDF
and detection functions, but, as the above DVM example illustrates, they may not contribute
information to the likelihood.
As with other distance sampling methods, edge effects exist, but these are taken account of
in estimation in two ways. Firstly, we model the reduction in the probability of detection as
M. J. Cox, D. L. Borchers, D. A. Demer, G. R. Cutter and A. S. Brierley
0
5
10
15
14
0.0
0.2
0.4
0.6
0.8
1.0
θ
Fig. 7. Estimate of the mean krill swarm biomass: correcting for size-biased angular detection probability
(the individual krill swarm biomass estimated by using the intercept of the regression of log(swarm biomass)
on the detection angle (see the text for details)): - - - - - - - , 95% confidence bounds
the deviation from vertical increases (see Fig. 5). This accounts for the reduced detectability of
swarms towards the edges of the swath. Secondly, we model the dependence of observed biomass on the deviation from the vertical (see Fig. 7). This should account for a reduced average
apparent size of swarms towards the edges of the pie slice (because swarms that are at greater
angles from the vertical will tend to have greater fractions of their biomass falling outside the
searched pie slice).
Analyses of SBE data typically assume perfect detectability of krill swarms in the beam
searched (e.g. Brierley et al. (1999) and Reiss et al. (2008), but see Demer et al. (1999)).
In our analysis of MBE data we have not made this assumption; how different would estimates be if we had made the assumption? We estimate that the swarm detection probability
has dropped by about 10% at 60 m and by about 20% at 100 m. The effect of this decline on
density and biomass estimates depends on the vertical distribution of krill. To investigate this
for our data, we reanalysed the survey data by assuming perfect detectability, i.e. p.r, θ/ = 1 for
Å
all (r, θ) in the swath searched (Fig. 2). In this case, the MLE of mean detection probability P̂ is
higher by about 20% and estimates of abundance N̂ s , biomass B̂ and density ρ̂ are lower by about
Å
12% (Table 4). The coefficient of variation of P̂ shrinks by a factor of about 3 when perfect
detectability is assumed, but this translates into a reduction of less than 2% in the coefficient
of variation of N̂ s , B̂ and ρ̂. If we assumed perfect detectability for folded MBE observation
angles, θ > 0:7 rad, then N̂ s -bias would be less than 12%. Although these biases are relatively
small, we recommend that the probability of detection is estimated when estimating abundance,
biomass or density from MBE data to avoid the bias.
Estimating Density of Antarctic Krill
15
Table 4. Model parameter and krill biomass variance
estimates assuming certain detection†
Parameter
Å
P̂
N̂ s
B̂ (tonnes)
ρ̂ (g m−2 )
Point
Coefficient-ofestimate
variation
estimate
0.54
1344.02
283.60
19.35
0.04
0.26
0.23
0.23
95%
confidence
interval
(0.49, 0.57)
(1101, 1984)
(201.99, 455.59)
(13.79, 31.09)
†Coefficients of variation were calculated by the non-parametric bootstrap (1000 replicates) with the transect as the
sampling unit; confidence intervals were calculated by using
the percentile method.
Although we have used the distance sampling methods that were developed here to estimate krill density and biomass, the methods are completely general for MBE surveys and are
applicable to MBE surveys of other species, including single animals.
Acknowledgements
We thank the Royal Society for funds enabling deployment of the MBE, Simrad USA,
particularly J. Condiotty, for the loan of the SM20, and D. Needham and M. Patterson of
Sea Technology Services for designing and constructing the MBE transducer mount. We are
grateful to the US Antarctic Marine Living Resources Program; the Master, officers and crew
of the RV Yuzhmorgeologiya and the personnel at the Cape Shirreff field station for logistical,
equipment and ship support. MJC was supported by a UK Natural Environment Research
Council doctoral studentship. This work was carried out in association with a National Science
Foundation funded (grant 06-OPP-33939) investigation of the Livingston Island near-shore
environment. A National Science Foundation grant was held by DAD and J. D. Warren of the
School of Marine and Atmospheric Sciences, Stony Brook University, USA. We are grateful to
E. Rexstad and T. Marques for both the provision of R code and helpful discussions of distance
sampling theory for MBE surveys. We thank A. Jenkins, T. S. Sessions and M. van den Berg for
skippering the near-shore boats and T. S. Sessions for providing invaluable expertise in the field.
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