Technical Reports of the Ins�tute of Cetacean Research (2021)

pp. 43–48

Technical Report (not peer reviewed)

An outline of the Density Surface Model for es�ma�ng abundance ofbaleen whales undertaken by the Ins�tute of Cetacean Research
Takashi Hakamada*Institute of Cetacean Research, 4–5 Toyomi-cho, Chuo-ku, Tokyo 104–0055, Japan*Contact e-mail: hakamada@cetacean.jp

ABSTRACTThe Ins�tute of Cetacean Research (ICR) is star�ng the use of the Density Surface Model (DSM), a model-based
approach, to es�mate abundance of large baleen whales. This is an alterna�ve approach to the design-based
approach used so far for the same purpose. This paper explains the concepts of DSM, and outlines the poten�al u�lity of this approach for the work on whale abundance undertaken by the ICR.

INTRODUCTION
The Ins�tute of Cetacean Research (ICR) conducts
regular sigh�ng surveys of large whales with the aimof es�ma�ng abundance in oceanic regions of interest.
The basic sigh�ng procedures for this purpose were described in Hakamada and Matsuoka (2017). Abundancehas been es�mated rou�nely by the ʻdesign-based approach,ʼ using the distance sampling method (Thomas
et al., 2010). Conven�onally, abundance is es�mated byequa�on (1) (Buckland et al., 1993; 2001).
AnE( s )2wL(1)

where,A is Area size of the survey area,n is the number of detected whale schools,
E(s) is the expected mean school size,
w is the eﬀec�ve half search width (esw), and
L is the searching distance.
The validity of the abundance es�mator based onequa�on (1) relies on the assump�on of a randomized survey design (Hedley and Buckland, 2004; Miller
et al., 2013). In the case that this assump�on is violated,
an abundance es�mator not requiring a randomized design is necessary.
Distribu�on of whales may be driven by environmental
covariates such as sea surface temperature (SST), depth
and salinity, each of which could vary in space. Rela�onships between environmental covariates and distribu�onof speciﬁc species is one of the interes�ng topics to inves�gate. Spa�al modelling of the whale distribu�on can beconstructed as a func�on of environmental covariates if
dependency of the covariates on density of whales is substan�al. Such constructed spa�al modelling can be used
to predict distribu�on in areas where the environmental
covariates are available. Inves�ga�on of environmental
covariates related to the distribu�on of the species canbe conducted by selec�ng covariates using sta�s�calcriterion.
The two aims of spa�al modelling are i) es�ma�on of
overall abundance in the area of interest (ʻmodel-based
approachʼ), and ii) inves�ga�on of the rela�onship between density/abundance and environmental covariates(Miller et al., 2013). Unlike conven�onal distance sampling (ʻdesign-based approachʼ), abundance es�matesobtained by spa�al modelling do not rely on the survey
design. Spa�al modelling can be applied to data obtained
from diﬀerent sources: pla�orms of opportunity suchas ferries, ﬁshing boats, incomplete surveys due to bad
weather or accident, and surveys that were designed
neither randomly nor systema�cally.
There are many kinds of spa�al modelling. Density Surface Model (DSM) (Miller et al., 2013) is one of them, andits use is star�ng to be u�lized at the ICR. The objec�ve of
this paper is to explain the basic concepts of DSM basedon Generalized Addi�ve Model (GAM) (Wood, 2006)
in the context of abundance es�mates of large baleen
whales. A secondary objec�ve of this paper is to outline
the future applica�ons of the DSM in the work on whale
abundance by the ICR.

GENERALIZED ADDITIVE MODEL (GAM)
DSM is expressed using the sta�s�cal model GAM. GAM
allows the assump�on that distribu�on of response vari43

Technical Reports of the Ins�tute of Cetacean Research (2021)

ables is an ʻexponen�al family.ʼ A distribu�on belongs tothe exponen�al family if its probability density func�on orprobability mass func�on can be expressed by the formula { yθ − b(θ )}+ c ( y ,φ ) f (θ , y ) = exp 
a
(
)
φ



(2)

where a, b and c are arbitrary func�ons, ϕ is an arbitraryscale parameter and ϑ is known as the canonical parameter of the distribu�on (Wood, 2006). Proper�es and examples of exponen�al families are provided in McCullagh
and Nelder (1989) and Wood (2006). In this context, GAM
can model variables that follow not only normal distribu�on but also Poisson, binomial, and other distribu�ontypes. For example, the number of counts, probability,
presence/absence can be used as a response variable in
GAM. A formula of GAM can be wri�en as:

E ( y ) = g −1 ( f1 ( x1 ) ++ f k ( xk ))(3)

where,y is the response variable,
E(y) is the expecta�on of y,
g−1 is the inverse of link func�on g,
fj(xj) is a smooth func�on of xj.
A spline is a special func�on deﬁned piecewise by polynomials, and is usually used as a smooth func�on in the
GAM. When fj(xj) is a linear func�on with respect to xj,
equa�on (3) becomes a Generalized Linear Model (GLM)(McCullagh and Nelder, 1989). In this sense, GAM can be
regarded as an extension of GLM.

Figure 1.

Histogram of rela�ve frequency of detec�ons by intervals of perpendicular distance and plot of detec�on func�on (bold curve). The red zone represents
the propor�on of animal detec�ons missed as compared to the animals in the surveyed area within T
n.miles from the track line, where T is trunca�on
distance. Example of the histogram and the detec�on func�on models are taken from Hakamada
et al. (2013).

There are four steps in construc�ng a DSM from line transect survey data.
1. Fi�ng detec�on func�on with perpendicular distance data and other covariates that would aﬀect
detectability of whale schools.
2. Fi�ng count models in each grid with environmental covariates data.
3. Predic�on of abundance in each grid in the study
area using the models constructed above.
4. Summa�on of the predicted abundance in the grid
over the area of interest.
The abundance es�mate in the study area is obtained
as follows.

distance. It is assumed that the detec�on probability
decreases as the perpendicular distance increases. Figure
1 shows a histogram of rela�ve frequency of detec�on by
perpendicular distance and es�mated detec�on func�on.
T is the trunca�on distance (i.e., detec�ons of perpendicular distance more than T are excluded from the analysis).
Among the animals within T n.miles from the track line,
the zone below the detec�on func�on represents animals
actually detected. The red zone represents the animals
missed. The detec�on func�on predicts probability foreach detec�on. The propor�on of the former is p, thedetec�on probability and propor�on of the la�er is 1-p.
By dividing the number of the detec�ons by the detec�onprobability p, the number of whales that fail to be detected can be taken into account (i.e., red zone in Figure 1).
The detec�on func�on derived from Mul� Covariate
Distance Sampling (MCDS) can be used in this method.
The advantage of using MCDS is that the eﬀect of covariates on sigh�ng condi�ons, such as Beaufort scale, windspeed, and visibility, on the detec�on func�on can beassessed. As men�oned later, the detec�on func�on withassump�on g(0)<1 can be used in this method.

Step 1The detec�on func�on represents the rela�onship between the detec�on probability and the perpendicular

Step 2
Figure 2 illustrates the division into grids to constructcount models. As shown in the ﬁgure, the track line sur-BASIC CONCEPT FOR THE CONSTRUCTION OF
DENSITY SURFACE MODEL (DSM)44

An outline of the Density Surface Model for estimating abundance of baleen whales undertaken by the Institute of Cetacean Research

Figure 2.

Illustra�on of the division of track line into grids to construct count models. Black circle indicates a detected whale school. Bold line indicates the track line surveyed. T is trunca�on distance. Lj is the length of
the track line surveyed in the j th grid. Area size of the j th grid is Aj =2LjT.

Figure 3.

Scheme to predict abundance es�mate in each grid.

veyed were divided into several sec�ons.
The j th grid has a width of 2T and length of Lj. Lj is
not necessarily constant. The area size of the j th grid is
Aj =2LjT. The area size should be chosen, such that it issmall enough to ensure that density in the grid can be
regarded as almost uniform. The number of the detectedschool in the j th grid is nj. The count model has covariatesthat could change in response to space and other factors(e.g., longitude, la�tude, SST, depth, salinity) as explanatory variables and the number of counts per unit area size
as a response variable.
For simplicity, it is assumed that the values of the explanatory variables xjs are available for any posi�on in
the area of interest. When ﬁ�ng the model, explanatory
variables in the grid are usually obtained by averaging all
observed values in the grid. One example of the countmodel can be expressed by:i=kE(n j ) = A j pj exp  b0 +fi ( xij ) 


i=1





(4)

where,
E(nj) is the expected the number of the detec�on in thegrid j,Aj is the area size of the grid j as an oﬀset,
pj is the averaged detec�on probability for detectedschools in the j th grid derived from the detec�on func�on constructed in step 1,
b0 is intercept,
k is the number of the explanatory variables, andfi(xij) is the i th smooth func�on of the i th covariate xij at
the j th grid.
When the detec�on func�on has covariates on sigh�ng

condi�on derived by MCDS, it can be considered that pj
may be diﬀerent among the grids.
Step 3
The grids for predic�on of abundance are deﬁned inorder to cover the study area of interest. The shape ofthe grid is usually almost a square. The size of the gridshould be small enough so that density can be regarded
as almost uniform. These grids are diﬀerent from thosedeﬁned in Step 2.
The expected numbers of detec�on (abundance es�mate) in the grid can be predicted by using equa�on(4), if Aj, pj and xij for all i are available (Figure 3). Aj is thearea size of the grid j and is clearly available. pj can bepredicted in all the grids if one of the two following condi�ons are sa�sﬁed. The ﬁrst is that the covariate of thedetec�on func�on derived in Step 1 is only the perpendicular distance (in this case, pj is constant for all grids).
The second is that the detec�on func�on has covariates
on sigh�ng condi�ons and the value of the covariates areavailable for all the grids. The explanatory variables in thegrid are usually obtained by averaging all observed values
in the grid. Thus, abundance es�mate in all the grids canbe predicted.
Step 4The predicted abundance es�mate in each grid are totaled to obtain the abundance es�mate for the wholearea of interest. When the shape of the area of interest
is complex, only a part of the grid belongs to the area forsome of the grids at the edge of the area. In such case,equa�on 4 can be applied to the intersect of the grid andthe area.
45

Technical Reports of the Ins�tute of Cetacean Research (2021)

numbers of grids constructed in Step 2 is larger than thenumbers of animals detected. Alterna�ve distribu�ons ofcount are, for example, quasi-Poisson, nega�ve binomial
and Tweedie distribu�on (Tweedie, 1984), which deal
with the higher propor�on of zero data. The latest version of R library dsm, which is so�ware that can conduct
DSM, can deal with quasi-Poisson, nega�ve binomial andTweedie distribu�on as the distribu�on of a response
variable (Miller et al., 2021).

Figure 4.

Illustra�ve example of a map for the predicted
abundance in each grid. Strong red indicates highdensity and strong blue indicates low density. Greenline indicates the boundary of the area of interest.

Figure 4 is an illustra�ve map for the predicted abundance in each grid. In this case, the model shows a tendency whereby the predicted abundance is ge�ng higher
from the right to the le�. For some of the grids, the wholeof the grid is not included in the area of interest. Predic�on can be conducted using the data in the intersect of
the grid and the area. One of the advantages of predic�ngabundance by the grids is that it is easy to calculate theabundance es�mate for any subset of the surveyed areaby summing the predicted abundance over the grids inthe subset.
Further aspects on DSM
One and two stage approaches
Explana�on in the previous sec�on is based on the twostage approach, which means ﬁ�ng DSM and detec�onmodel at each stage. A one-stage approach means ﬁt�ng both models simultaneously (Royle et al., 2004).
The disadvantage of the one-stage approach is that thecomputa�on to es�mate and check is more diﬃcult thanin the two-stage approach, because both steps must be
conducted at once. The disadvantage of a two-stage approach is that to es�mate uncertainty for the abundancees�mate, the uncertain�es in the detec�on func�onand the spa�al model must be combined appropriately(Miller et al., 2013).
Distribution of response variable
In the case of count models, Poisson distribu�on is usually assumed as the distribu�on of the count. In manycases, there is a high propor�on of the grids with zero
count (i.e., there is no sigh�ngs in the grid), because the46

Variance estimation
In previous studies on abundance es�mates derived from
DSM, the variance was es�mated using parametric bootstrap (Hedley et al., 1999; Hedley and Buckland 2004).
When the detec�on func�on model and the count model
are independent, the variance of the abundance can beapproximated as the sum of variance due to detec�onfunc�on and variance due to count model using delta
method (Seber, 1982). When the detec�on func�on andthe count model are not independent, the approxima�onby the delta method cannot be applied to es�mate thevariance of abundance. In this case, variance propaga�on
can be applied to es�mate the variance of the abundance. Details of the variance propaga�on are provided
in Bravington et al. (2021).
Evaluation of extrapolation using DSM
When applying DSM to areas outside of the reference data
(i.e., data used to construct the model), a�en�on shouldbe paid to the extrapola�on of DSM (Miller et al., 2013).
For example, when the DSM that has SST as a covariate isﬁ�ed using data from warm waters in a temperate zone,
predic�ons for cold waters in the sub-arc�c by the model
are unlikely to be reliable due to diﬀerence in the rangeof SST. Bouchet et al (2020a) reviewed several quan�ta�ve extrapola�on diagnos�cs and recommended to use
two of them, principally, the percentage of data nearby,%N (King and Zeng, 2007) and the Extrapola�on Detec�on, ExDet (Mesgaran et al., 2014), as standard tools forassessing extrapola�on in abundance models. %N is ametric that represents how close the data for extrapola�on and the reference data are. The higher the %N is,
the more reliable the extrapola�on is. ExDet is a metricthat characterizes both univariate and combinatorial extrapola�on and provides geographical distribu�on of themost inﬂuen�al covariate among the covariates (Bouchet
et al., 2020a). Assessment of extrapola�on using the two
metrics can be implemented by dsmextra library of R
(Bouchet et al., 2020b).

An outline of the Density Surface Model for estimating abundance of baleen whales undertaken by the Institute of Cetacean Research

POTENTIAL UTILITY OF THE DSM FOR THE WORKON WHALE ABUNDANCE BY THE INSTITUTE OFCETACEAN RESEARCH
The model-based approach can be applied to the case of
Antarc�c minke whales and common minke whales in thewestern North Paciﬁc. These are two of the target species
for abundance research by the ICR.
In the case of Antarc�c minke whales, the abundancees�ma�on in polynyas is problema�c. The pack ice prevents the research vessels entering the polynyas. The
shape of the polynyas is o�en complex and narrow,
therefore a random and/or systema�c survey design cannot be realized even if pla�orms other than the vessel(e.g., drone) are used for the survey. In such cases, thedata obtained can be used to es�mate abundance basedon DSM.
The common minke whales in the western North
Paciﬁc are distributed in the Okhotsk Sea, the western
North Paciﬁc Ocean and the Sea of Japan. Some areasof the distribu�on of this species cannot be surveyed,because they are in the Exclusive Economic Zone (EEZ) of
third countries, from which permits are diﬃcult to obtain.
This situa�on results in underes�ma�on of the abundance because not all areas of distribu�on of the species
or stock are covered. If some environmental data in theunsurveyed areas are available, extrapola�on by modelbased approach can be applied. This study would beambi�ous and challenging, because it is diﬃcult to obtainreliable DSM given that the ranges of environmental covariates in areas where the DSM was extrapolated would
be expected to be diﬀerent from the reference data (i.e.,the data used to construct the model). However, the reliability of the extrapola�on can be evaluated using theindices men�oned above. By assessing condi�ons underwhich models are likely to fail or succeed in extrapola�onthrough this exercise, a be�er understanding of distribu�on pa�erns and their underlying drivers for this species
may be obtained (Yates et al., 2018).
The latest version of R library dsm can deal with detec�on func�on derived from mark-recapture distancesampling (MRDS) that allows g(0)<1 (Miller et al., 2021).
Concept and example of such detec�on func�on derived
from the MRDS are provided in Takahashi (2019). From
this update, DSM derives abundance es�mates that consider g(0) es�mate. This is par�cularly relevant for species that cannot be detected due to their behavior andwhose distribu�on is driven by environmental covariates
varying in space. DSM that allows g(0)<1 can be applied
to whales, such as common minke whales and Antarc�cminke whales, whose g(0) es�mates have been shown tobe less than 1 in previous studies (Okamura et al., 2010;
Okamura and Kitakado, 2012).

ACKNOWLEDGEMENTS
I thank Luis A. Pastene (ICR) for his assistance in preparing
this manuscript.

REFERENCES
Bouchet, P.J., Miller, D.L., Roberts, J.J., Mannocci, L., Harris,
C.M. and Thomas, L. 2020a. From here and now to there
and then: Prac�cal recommenda�ons for extrapola�ng cetacean density surface models to novel condi�ons. Technicalreport 2019-01 v2.0, Centre for Research into Ecological &Environmental Modelling (CREEM). University of St Andrews,
St Andrews. 59 pp.
Bouchet, P.J., Miller, D.L., Roberts, J.J, Mannocci, L., Harris, C.L.
and Thomas, L. 2020b. dsmextra: Extrapola�on assessment
tools for density surface models. Methods in Ecology and
Evolution 11: 1464–1469.
Bravington, M.V., Miller, D.L. and Hedley, S.L. 2021. Variancepropaga�on for density surface models. JABES 26: 306–323.
Buckland, S.T., Anderson, D.R., Burnham, K.P. and Laake, J.L.
1993. Distance Sampling: Estimating Abundance of Biological
Populations. Chapman and Hall, London. 446 pp.
Buckland, S.T., Anderson, D.R., Burnham, K.P., Laake, J.L.,
Borchers, D.L. and Thomas, L. 2001. Introduction to DistanceSampling. Oxford University Press, Oxford. 595 pp.
Hakamada, T. and Matsuoka, K. 2017. Sigh�ng survey procedures for abundance es�mates of large whales in JARPA and
JARPAII, and results for Antarc�c minke whales. Technical
Reports of the Institute of Cetacean Research (TEREP-ICR) No.
1: 28–36.
Hakamada, T., Matsuoka, K., Nishiwaki, S. and Kitakado, T. 2013.
Abundance es�mates and trends for Antarc�c minke whales(Balaenoptera bonaerensis) in Antarc�c Areas IV and V basedon JARPA sigh�ng data. J. Cetacean Res. Manage. 13(2):123–151.
Hedley, S.L., Buckland, S.T. and Borchers, D.L. 1999. Spa�almodelling from line transect data. J. Cetacean Res. Manage.
1: 255–264.
Hedley, S.L. and Buckland, S.T. 2004. Spa�al models for line transect sampling. JABES 9: 181–199.
King, G. and Zeng, L. 2007. When can history be our guide? Thepi�alls of counterfactual inference. Int. Stud. Q. 51: 183–210.
McCullagh, P. and Nelder, J.A. 1989. Monographs on Statistics
and Applied Probability. 37. Generalized linear model, 2ndedition. Chapman and Hall, London. 511 pp.
Mesgaran, M.B., Cousens, R.D. and Webber, B.L. 2014. Here be
dragons: A tool for quan�fying novelty due to covariate range
and correla�on change when projec�ng species distribu�onmodels. Divers. Distrib. 20: 1147–1159.
Miller, D.L., Burt, M.L., Lexsdat, E.A. and Thomas, L. 2013.
47

Technical Reports of the Ins�tute of Cetacean Research (2021)
Spa�al models for distance sampling data: recent developments and future direc�ons. Methods in Ecology and
Evolution 4: 1001–1010.
Miller, D.L., Rexstad, E.A., Burt, M.L., Bravington, M.V. andHedley, S.L. 2021. dsm: Density surface modelling of distancesampling data. Ver. 2.3.1. URL h�p://github.com/dill/dsm.
Okamura, H., Miyashita, T. and Kitakado, T. 2010. g(0) es�mates
for western North Paciﬁc common minke whales. Paper
SC/62/NPM9 presented to the IWC Scien�ﬁc Commi�ee,June 2010 (unpublished). 7 pp. [Available from the IWC
Secretariat].
Okamura, H. and Kitakado, T. 2012. Abundance es�matesof Antarc�c minke whales using the OK method. Paper
SC/64/IA2 presented to the IWC Scien�ﬁc Commi�ee,June 2012 (unpublished). 24 pp. [Available from the IWC
Secretariat].
Royle, J., Dawson, D. and Bates, S. 2004. Modeling abundance
eﬀects in distance sampling. Ecology 85: 1591–1597.
Seber, G.A.F. 1982. The estimation of animal abundance andrelated parameters. Macmillian, New York. 654 pp.
Takahashi, M. 2019. A note on g(0) es�mates derived from vessel-based sigh�ng surveys. Technical Reports of the Instituteof Cetacean Research (TEREP-ICR) No. 3: 14–20.
Thomas, L., Buckland, S.T., Rexstad, E.A., Laake, J.L., Strindberg,S., Hedley, S.L., Bishop, J.R., Marques, T.A. and Burnham,

48

K.P. 2010. Distance so�ware: design and analysis of distancesampling surveys for es�ma�ng popula�on size. J. Appl. Ecol.
47: 5–14.
Tweedie, M.C.K. 1984. An index which dis�nguishes between
some important exponen�al families. pp. 579–604. In: J.K.
Ghosh and J. Roy (eds.). Statistics: Applications and New
Directions. Proceedings of the Indian Statistical InstituteGolden Jubilee International Conference. Indian Sta�s�cal
Ins�tute, Calcu�a. 609 pp.
Wood, S.N. 2006. Generalized Additive Models: An introduction
with R. Chapman and Hall/CRC, Boca Raton, FL, USA. 391 pp.
Yates, K.L., Bouchet, P.J., Caley, M.J., Mengersen, K., Randin,C.F., Parnell, S., Fielding, A.H., Bamford, A.J., Ban, S., Barbosa,A.M., Dormann, C.F., Elith, J., Embling, C.B., Ervin, G.N., Fisher,R., Gould, S., Graf, R.F., Gregr, E.J., Halpin, P.N., Heikkinen,
R.K., Heinänen, S., Jones, A.R., Krishnakumar, P.K., Lauria,
V., Lozano-Montes, H., Mannocci, L., Mellin, C., Mesgaran,
M.B., Moreno-Amat, E., Mormede, S., Novaczek, E., Oppel, S.,
Ortuño Crespo, G., Peterson, A.T., Rapacciuolo, G., Roberts,
J.J., Ross, R.E., Scales, K.L., Schoeman, D., Snelgrove, P.,
Sundblad, G., Thuiller, W., Torres, L.G., Verbruggen, H., Wang,
L., Wenger, S., Whi�ngham, M.J., Zharikov, Y., Zurell, D. andSequeira, A.M.M. 2018. Outstanding challenges in the transferability of ecological models. Trends Ecol. Evol. 33: 790–802.