Population Estimates From Aerial Photographic Surveys of Naturally and Variably Marked Bowhead Whales

Transcription

1 Population Estimates From Aerial Photographic Surveys of Naturally and Variably Marked Bowhead Whales Tore SCHWEDER,DinaraSADYKOVA, DavidRUGH, and William KOSKI Abundance, mortality, and population growth of bowhead whales (Balaena mysticetus) are estimated from captures of 4,894 putatively different individuals obtained from 10 years of systematic photographic surveys conducted during the spring migration when most of the Bering Chukchi Beaufort population of bowheads migrates past Point Barrow, Alaska. A stringent matching protocol designed to prevent false positive matches of the naturally, but variably marked individuals, led to 42 resightings between years. The flip side of this stringency is a presence of false negatives, i.e., some true recaptures are not recognized as such. The problem of false negatives is addressed by modeling the capture process and the matching process. The captures of an individual are assumed to follow a Poisson process with intensity depending stochastically on the individual whale and on the year. The probability of successfully matching a capture to a previous capture is estimated by logistic regression on the degree of marking and image quality. Individuals are recruited by the Pella Tomlinson population model, and their mortality rate is assumed to be constant. The point estimate of yearly growth rate is 3.2%, and bowhead abundance in 2001 is estimated to be 8,250, similar to previous estimates. Key Words: Abundance; Bootstrap; Capture recapture; Confidence curve; Event history; False negatives; Photo-identification. 1. INTRODUCTION Images of photographically captured bowhead whales are compared to previously captured images, with respect to natural marks, for the purpose of constructing capture histo- Tore Schweder is Professor, Department of Economics, University of Oslo, P.O. Box 1095, Blindern, 0317 Oslo, Norway and Centre for Ecological and Evolutionary Synthesis, Biological Department, University of Oslo, P.O. Box 1066, Blindern, 0316 Oslo, Norway ( tore.schweder@econ.uio.no). Dinara Sadykova is Ph.D. Student, Centre for Ecological and Evolutionary Synthesis, Biological Department, University of Oslo, P.O. Box 1066, Blindern, 0316 Oslo, Norway ( dinara.sadykova@bio.uio.no). David Rugh is Wildlife Biologist, National Marine Mammal Laboratory, NOAA Fisheries, 7600 Sand Point Way, NE, Seattle, WA 98115, USA ( bkoski@lgl.com). William Koski is Senior Environmental Scientist, LGL Limited environmental research associates, 22 Fisher Street, P.O. Box 280, King City, Ontario, L7B 1A6 Canada ( bkoski@lgl.com) American Statistical Association and the International Biometric Society Journal of Agricultural, Biological, and Environmental Statistics, Volume 15, Number 1, Pages 1 19 DOI: /s

2 2 T. SCHWEDER ET AL. ries for individuals. The matching process results in some definite matches, but due to the variability in degree of marking and also that of image quality, some of the recorded nonmatches will be false negatives. Many of the constructed capture histories will, therefore, be incomplete, and the same individual might be represented by several recorded capture histories. Our aim is to tackle the problem of false negatives in our capture recapture study using data from 10 years ( , plus 2003) of systematic photo surveys designed to estimate abundance and demographic parameters of bowhead whales in Alaskan waters. The Bering Chukchi Beaufort (BCB) bowhead whale population size and rate of increase has been assessed by periodic ice-based visual surveys assisted by passive acoustics (George et al. 2004; Zeh and Punt 2005). The most recent estimate from 2001 is of 10,545 individuals, with a CV of 0.13 with a rate of increase of 3.4% (95% CI = 1.7% 5%). With changes in sea ice conditions due to global warming, there is concern that the icebased census will not be feasible in the future. The question we address in this article is whether data from the photo surveys are in accordance with other data for the BCB bowhead whales, and whether results from these surveys might contribute to, or replace, the ice-based census during future population assessments relevant for management. Our model and method might also be of interest in other capture recapture studies where actual recapture rates are uncertain, but any recognized recaptures are considered true. BCB bowheads winter in the Bering Sea and migrate through the Chukchi and western Beaufort seas to summering areas in the eastern Beaufort Sea and Amundsen Gulf (Moore and Reeves 1993). Typically, bowheads pass Barrow, Alaska, from mid-april to early June (George et al. 2004), and during most years from 1984 to 1994 and 2003, systematic aerial photography surveys were conducted near Barrow during the whales spring migration (Angliss et al. 1995; Koski et al. 2006). Although listed as endangered under the U.S. Endangered Species Act (Shelden et al. 2001), this population is subject to aboriginal subsistence whaling managed by the International Whaling Commission, with an average of 56 bowheads per year taken in recent years. In previous analyses of the photographic data (Rugh 1990; da Silva et al. 2000; Zeh et al. 2002; Schweder 2003), individual whales were categorized as either marked or unmarked, or as in da-silva, Gomes, and Stradioto (2007), as highly marked, moderately marked, or unmarked, and matching was regarded as conclusive for marked whales. Assuming that the matching is correct, particularly that two unmatched captures always refer to different individuals, is an over-simplification of the process because it is known that not all aerial images of whales provide perfect viewing of their dorsal surfaces, and not all whales are sufficiently marked to be identifiable. When photographs are manually processed, each image representing a capture is scored for degree of marking and quality of image at rostrum, mid-back, lower back, and fluke (Rugh et al. 1998). Past studies ignored poorly marked whales as part of the marked population and discarded images scored as the poorest quality. Here we model the probability of correctly identifying a recaptured individual as a function of the scores. This incorporates more of the data into the analyses and eliminates the need to treat marked and unmarked whales separately in models used to estimate population size.

3 PHOT-SURVEYS OF NATURALLY MARKED WHALES 3 The collection of bowhead whales has grown steadily across several decades. Until recently it was not impractical to have all images matched manually following a filtering of the datasets by categorizing images according to the amount of white on rostral or peduncle areas. By the time there were large additions to the collection in 2003 and 2004, there was increased incentive to develop a computerized matching program for efficient searching for matches between images. Although a program has been developed, it is currently undergoing testing and is not ready for reliable application to the full collection. The capture histories considered here are thus based on images that were matched by visual inspection according to a specific protocol (Rugh et al. 1998). Each captured individual is exposed to two competing options: of dying or being subsequently captured (whether or not it is recognized in subsequent photographs). In our event history model, individuals are recruited at the beginning of each year in proportion to abundance. They die at constant mortality intensities, and they are captured during surveys according to a Poisson process, with intensity depending stochastically on the individual and on the year. Abundance, net population growth rate, mortality intensity, Pella Tomlinson parameters, capture intensity parameters, and logistic regression parameters for successful matching are estimated by maximum likelihood. These estimates are biased due to the approximate nature of the likelihood. A simulation experiment is, therefore, conducted to correct for inherent potential biases and to integrate the various sources of variability, allowing the construction of confidence curves and confidence intervals for abundance, population growth, and individual mortality. 2. MATERIAL Data on bowhead images were obtained from the National Marine Mammal Laboratory, National Oceanic and Atmospheric Administration (NOAA) Fisheries Service, and LGL Limited, who share the data management. The file contains data on 6,800 bowhead images of acceptable quality that were obtained from photographs taken near Point Barrow during the bowhead spring migration (calves are excluded from consideration here). Photographic surveys were conducted at Barrow in the spring of each year , , 1994, and Captures were screened for matches within the first period ( ) and within 2003, but not between the first period and Among the 4,894 captured putative individual whales, there were 42 recognized recaptures between years (Rugh et al. 2008). The record for each putative whale scores the degree of marking for each of four zones (rostrum, mid-back, lower back, and fluke) on an ordinal six-point scale from highly marked to unmarked, and undeterminable; and the visibility of each of the four zones is categorized with respect to five ordinal categories of image quality from excellent to poor and missing values (null) according to Rugh et al. (1998). We converted the ordinal scores for degree of marking to 5, 4, 3, 2, 1, 0 and for image quality to 4, 3, 2, 1, 0, where undeterminable or missing also is scored 0. The record also contains location, year, date, and time of capture and an ID-number which is repeated if the capture is regarded as a resighting of a previously recorded individual. The time spent on effort is also recorded for each survey day. Mean effort was 2.47 hr per day for days on which aerial surveys were

4 4 T. SCHWEDER ET AL. Table 1. Survey effort (hours on primary effort), number of captures, number of putative individuals that were recaptured later (First), and number of putative captured for the second time (Second). Year Effort Captures First Second NA NA Total , NA = not available. conducted, excluding 1984 when hours on effort were not recorded. Survey effort (hours on primary effort), number of captures, number of whales that were recaptured later seen in the first sighting, and number of whales seen in the second sighting (recaptures) are indicated in Table 1, relative to year. Survey effort for 1984 is estimated from the number of survey days (10 days between May 4 and 13) using the overall average of 2.47 hr per day. 3. METHODS We first develop a stochastic model for the population process, the capture process, and the matching. This leads to an approximate likelihood function. We use simulations to correct for various sources of bias and to integrate the various sources of variability. Our inference will, therefore, be based on parametric bootstrapping MODEL Capture recapture surveys with uncertain matching are complex processes. Since matching is done with great care in order to avoid false positive matches (with the option of multiple observers checking a match), we will assume that there are no such errors. There are, however, false negative matches generating ghost capture histories, and at the same time, incomplete capture histories. For each capture, a covariate vector is recorded characterizing degree of marking and quality of image for the four body zones. The data also indicate whether the capture is of a cow (adult whale) with a calf (<1 year old), and for some captures the body length of the individual is estimated fairly precisely through photogrammetry. We use degree of marking and quality of image as covariates in a logistic regression for the probability of a correct match in a subsequent capture of the individual. This enables us to simulate the matching process, and thus, to handle the problem with false negative matches.

5 PHOT-SURVEYS OF NATURALLY MARKED WHALES 5 Finally, bowhead whales are recruited to the surveyed population of age 1+, and are assumed to be subject to a constant mortality rate at all ages. The population dynamics is assumed to be of the deterministic Pella Tomlinson type for the age-lumped population. The abundance in year y, N y is thus assumed to grow according to ( ) z ) Ny N y+1 = N y + ρn y (1, (3.1) K where ρ,k,z, respectively, the net growth rate at low population level, the carrying capacity, and the shape parameter, are parameters to be estimated. The survey data used in this study are inadequate for estimating K and z with any precision. We do, however, entertain the Pella Tomlinson model in the interest of generality and for enabling an integrated analysis of photographic survey data together with other types of data, as in Brandon and Wade (2006). We also fit a model with simple exponential population growth, i.e., K = in Equation (3.1). Matches from a new survey are in reality identified for each capture by searching through the previous captures. Despite this back-searching, we shall think of matching as a search forward in time. Unmatched captures are in focus one at the time in chronological order. For each capture, a search for its next match is done by comparing it to later unmatched captures chronologically ordered. Let x be the covariate vector in a logistic regression for the probability of correctly matching if the new capture is the same individual. The predictor is only based on characteristics of the first capture in the pair, and whether the new capture is in the same survey. This way the model has Markovian structure, allowing a likelihood function to be established. We consider 10 yearly surveys, see Table 1. We will use survey and year interchangeably. A capture is compared to subsequent captures in the same survey and in later surveys. The initial search for matches is done in the same way in both cases, but within a survey position at capture and also body length (if available) is considered. Since there is more information available for matching within than between surveys, an indicator variable for within survey is included in the linear logistic regression model for the probability of a true recapture successfully being matched. This part of the model is called the logistic regression although the logistic parameter is estimated along with the remaining parameters by maximizing the approximate likelihood found in Equation (3.3), below. With Q j denoting the quality score and M j the score for degree of marking for body zone j, and with W denoting an indicator for the comparison being within survey, the linear predictor in the logistic regression is η = β j=1 (β j M j + β 4+j Q j ) + β 9 W = x β in obvious notation. A pair of captures of the same individual are successfully matched with probability p = p(x; β) = 1/(1 + exp( η)). Bowhead whales acquire natural marks at birth, and from contact with ice, killer whales, ships, or other objects, such as the sea floor (Rugh, Braham, and Miller 1992; George et al. 1994). To assume that the array of marks is static is incorrect because marks accumulate on each whale s dorsal surface as their age increases (Davis, Koski, and Miller 1983). The rate of accumulation of marks appears to be slow, but has not been quantified. There are clearly also measurement errors in recorded degrees of marking. We will not attempt to account for measurement errors in scoring, but see Section 6.4 below.

6 6 T. SCHWEDER ET AL. As the surveys progress, the capture histories are dynamically constructed. Assuming that possible matching of a new capture is based on the previous capture/recapture in the history, we let the covariate vector x for the logistic regression for correct matching be as observed in the last capture in the history to this point. The rate at which an individual is captured depends on various factors. The speed of passage through the survey area might vary from year to year according to oceanographic conditions, affecting many individuals, and the weather may preclude surveying for one or more days. The capture intensity should thus vary from survey to survey. It might also vary across individuals within years, but will an individual-specific component of the capture intensity be correlated across years? Individuals captured in several surveys seem to find their temporal rank in the spring migration past Barrow independently from survey to survey. This is seen by temporally ranking the captures within surveys, and analyzing the observed relative ranks for recaptured individuals. The year to year correlation in relative rank is estimated to be 0.1, and found to be not significantly different from zero (p-value 0.5) (Sadykova and Schweder 2009). We will, therefore, assume that the random component in the capture intensity specific to an individual is independent across years. The process of true captures of an individual i in year y is assumed to depend stochastically also on y, and is modeled as a Poisson process with intensity λ iy = τ iy λ y, where the random effects specific to the individual {τ iy } are independent and gamma distributed with density Ɣ(α) αα τ α 1 exp( ατ) to make Eτ = 1, and the year effects {λ y } are independent and lognormally distributed. The individual effects are modeled by the gamma distribution to allow for analytic integration and thus for computational ease APPROXIMATE LIKELIHOOD The likelihood is based on the assumption of all recorded capture histories only containing captures of one individual, while the same individual might be represented in more than one capture history. The term capture history refers to recorded capture histories. Since computational speed is desirable to allow many replicate simulations, approximations to the likelihood are done. Since we use parametric bootstrapping for inference, the plug-in principle (Efron and Tibshirani 1993) applies to the maximum (approximate) likelihood estimator. The likelihood is developed by first conditioning on the mean capture intensities within years λ 1,...,λ T, where T is the last survey year. This conditional likelihood is then integrated over these T random effects. Consider a capture history with an initial capture in survey year s. All recaptures in the history are captures of this individual (no false positives), but some true captures might not have been recognized as recaptures, and are lost from the capture history. Captures in a capture history subsequent to the initial capture are called recaptures. Many capture histories turn out to be solitary, i.e., containing only an initial capture. Assume for a moment that there is no within survey effect on the matching probability p, the numbers of captures over the exposures h y are conditionally independent, and have

7 PHOT-SURVEYS OF NATURALLY MARKED WHALES 7 probability distribution ( ) Ɣ(α + n) n ( ) μy α α f(n; μ y ) = (n 0). Ɣ(α)n! α + μ y α + μ y This negative binomial distribution of the number of (matched) recaptures n y is obtained by integrating out the gamma-distributed individual effect τ in the capture intensity in survey year y when the individual is assumed alive. The expected number of recaptures is μ y = λ y h y p, where p is the matching probability of the individual, and h y is the survey effort the individual is exposed to after the initial capture in the initial year of capture, while h y is total survey effort in subsequent years. The individual behind the capture history in focus is assumed to survive the initial year s of capture and to be alive through a future survey d years ahead with probability q d.the probability that the individual is alive in the survey in year y s, but dead in later surveys π sy is, thus, a function of annual survival probability q. With i indexing capture histories, the terms n y, λ y, h y, and p could be indexed by i above. The likelihood of obtaining the observed recaptures in capture history i, conditioning on the initial capture and the stochastic year effects in the capture intensity, is thus, L i = T y=s i π s i y y t=s i f(n i t ; μi t ). Note that the within year recaptures are included in L i. There are altogether 57 within year recaptures. These likelihood components are conditionally independent given the capture intensities over the years. To ease computations, we further approximate the likelihoods of the many histories with a solitary capture in year s (no recaptures) by π ss f(0; λ s ph s /2) + π sy y>s y f(0; λ t ph t ), (3.2) where the matching probability is set to p = p(x) and the mean is taken over solitary capture histories. The model is also fitted without this approximation (Table 2). A modification is needed in these formulas when n i t > 1 and when the matching probability p depends on the comparison being within a survey. Due to the Markovian structure of the capture and matching process, the correct likelihood is found by successive conditioning. We will also need the likelihood of the number of initial captures in all the putative capture histories. Still conditioning on the random year effects in the capture intensity, an individual is captured a stochastic number of times e iy in year y. These numbers are conditionally independent and negative binomially distributed with mean and variance, respectively, Ee iy = λ y h y and var(e iy ) = λ y h y (1 + λ y h y /α). The number of captures for all the N y individuals is thus by the Central limit theorem nearly normally distributed. Due to the low matching probability, nearly all the captures will generate new histories. An approximate conditional likelihood for the number of new capture histories m y starting in t=s

8 8 T. SCHWEDER ET AL. Table 2. Maximum likelihood estimates (MLE) and bias corrected estimates (BCE) from parametric bootstrapping. Models 1, 2, and 4 use the approximation of Equation (3.2) and are nested, while Model 3 does not utilize this approximation but is otherwise identical to Model 4. Deviance is twice the log-likelihood ratio relative to Model 1. Model: MLE MLE BCE MLE MLE BCE Abundance ,929 3,950 5,080 4,001 3,952 5,011 Abundance ,021 6,529 8,630 6,613 6,532 8,250 Abundance ,462 7,796 10,115 7,896 7,800 10,020 Growt ρ Mortality log(q) λ = median(λ y ) σ 2 = var(log(λ y )) α K z Deviance year y is thus, L y,init = ( 1 Ny λ y h y (1 + λ y h y /α) exp 1 2 (m y N y λ y h y ) 2 N y λ y h y (1 + λ y h y /α) Consider the set M y of capture histories with initial capture in year y. Conditional on the yearly capture intensity, these capture histories contribute a likelihood component L y,init i M y L i. These components are conditionally independent across years, and integrating out the random year effects in the capture intensities, the approximate likelihood is L = L i L y,init g(λ y )dλ y, (3.3) 0 0 y i where g is the lognormal density for λ y with parameters λ = median(λ y ) and σ 2 = var log(λ y ). Furthermore, this likelihood is a function of the shape parameter in the gamma distribution α, the 10-dimensional logistic regression parameter for matching success β, the survival probability q, the abundance in the first year of survey N 1, and the parameters ρ, z, and K governing the population process. Since images have not been searched for matches between 2003 and previous years, a separate log-likelihood component based on data from 2003 is added to the likelihood based on data from ) INFERENCE Due to the nonlinearities in the model and the approximations in the likelihood, maximum likelihood estimator of the vector of the 18 free parameters is expected to be biased and to have nonnormal sampling distributions. Therefore, a simulation experiment is carried out, and inferences in the form of confidence curves are constructed from the maximum likelihood estimates and their bootstrap distributions.

9 PHOT-SURVEYS OF NATURALLY MARKED WHALES 9 The confidence curve introduced by Birnbaum (1961) provides a convenient graphical display of a confidence distribution and confidence intervals at various levels. When C(ψ) is the cumulative confidence distribution function for a scalar parameter ψ (Schweder and Hjort 2002), N(ψ)= 2 C(ψ) 0.5 is the confidence curve. The interval between the two branches of the confidence curve at height cl is actually a confidence interval of confidence level cl. So, for cl = 0.95, the 95% confidence interval is read off the graph as the interval between the two branches at vertical level For cl close to 0, the interval becomes very short and is typically only a point at height 0, and the confidence curve is a funnel plot pointing at the median of the confidence distribution. This focal point is a point estimator which is median unbiased. There are several methods for obtaining confidence distributions from parametric or nonparametric bootstrapping (Efron and Tibshirani 1993). We shall only consider the (acceleration and) bias corrected bootstrap method introduced by Efron (Efron 1987; Schweder and Hjort 2002), but assuming no acceleration. The method is based on the assumption that there exists an increasing transformation φ that would have allowed us to consider instead of ψ and ˆψ transformation γ = φ(ψ) and ˆγ = φ( ˆψ), respectively, such that (γ ˆγ) b N(0, 1), where b is a bias parameter on the transformed scale. The unknown transformation φ is essentially estimated from the (parametric) bootstrap distribution of the estimator with cdf Ĝ. Let,as usual, be the standard normal cdf. The confidence distribution for ψ that we use is Ĉ(ψ) = ( 1 (Ĝ(ψ)) 2 1 (Ĝ( ˆψ))) (Efron 1987; Schweder and Hjort 2002), and the confidence curve follows. 4. SIMULATION We simulated the bowhead population over the 20 years and also the photo surveys with search effort as given in Table 1. The initial population size referring to year 1984 is N 1, and the population size in year y + 1 (counted from 1984) is given by Equation (3.1). The maximum likelihood estimates were used for initial population size, the annual rate of population increase, the mortality intensity, the carrying capacity and the Pella Tomlinson shape parameter, the capture intensity parameters, and the logistic regression parameter for the probability of successful matching. These are the MLE for Model 4 in Table 2 and the MLE for β given in Section 5 below. Since α was estimated to practically infinity, we simulate without individual-specific heterogeneity in the capture intensity. There is, however, still individual heterogeneity in the matching probability. The survey results for individuals of the simulated population were simulated one by one, after having drawn the capture intensity for each survey from the log-normal distribution. Each individual was assigned a geometrically distributed survival time and was removed from the population accordingly. Its scores for degree of marking and for the quality of the image respective of each zone were selected randomly from the original data, and its matching probabilities (between and within surveys) were calculated from the fitted logistic model. Poisson distributed numbers of captures were then generated for each survey in which the individual was present, and one of these was randomly chosen as an

10 10 T. SCHWEDER ET AL. observed capture of the individual. The remaining sequence of captures was then thinned by independent Bernoulli trials according to the matching probabilities of the individual. Each of the thinned out captures (the false negative matches) yielded an observed solitary capture history (no recaptures). The captures remaining after thinning (the true positives), together with the randomly selected capture, yielded an observed capture history. The resulting set of simulated observed capture histories was finally summarized by the number of solitary capture histories by survey and the set of histories with recaptures. The model was fitted to simulated data exactly in the same way as it was fitted to the observed data (see below). This was done for 1000 simulated replicates of the data. For each replicate, point estimates were calculated for all the parameters, including the derived abundance parameter N y for year y counted from RESULTS The model was fitted by way of the computer package AD Model Builder ( admb-project.org/) using the approximate likelihood in Equation (3.3), where numerical integration is done by the Laplace transformation (Skaug and Fournier 2006). Maximum likelihood estimates, median unbiased estimates, and 95% confidence intervals are found in Tables 2 and 3. Also given, for the parameters of primary interest, are the rough 95% confidence intervals found from the likelihood profiles by chi-square contouring. We give results for four models (Table 2). Models 1, 2, and 4 are nested, while Model 3 does not rely on the approximation of Equation (3.2). Of the nested models, the simple Model 1 fits the data significantly worst, and we conclude that capture intensity varies significantly over the years. When choosing between Models 2 and 4, it is clear from Tables 2 and 3 that the inclusion of the parameters K and z representing density dependence in the population dynamics does not improve the fit to any degree. From the similarity in estimates and standard errors, the inclusion of these parameters seems, however, not to inflict any loss in estimation precision (Tables 2 and 3). In the interest of preparing for an integrated assessment of the BCB bowhead whales based on the Pella Tomlinson model, where harvest data, other survey data, age data, and other Table 3. Standard errors from the Hessian of the log-likelihood (SE) and 95% confidence intervals from bootstrapping (95% CI) and from profile likelihoods (95% CI, LP) for parameters of interest that are acceptably estimated. The models are as in Table 2. Model: SE SE SE SE 95% CI 95% CI, LP Abundance (2001, 9,298) (2092, 6,497) Abundance 2001 (3,150, 15,450) Abundance 2007 (3,718, 18,400) Growt ρ (0.010, 0.084) (0.005, 0.048) Mortality log(q) (0.013, 0.024) (0.011, 0.023) λ = median(λ y ) (0.0008, ) σ 2 = var(log(λ y )) (0.003, 0.22)

11 PHOT-SURVEYS OF NATURALLY MARKED WHALES 11 relevant data are analyzed together with the photographic survey data, we have a weak preference for Model 4. Model 3 has the best likelihood and would have been selected by any standard information criterion such as AIC. Despite the better likelihood for Model 3 due to the better handling of the solitary capture histories by avoiding approximation in Equation (3.2), we choose to use Model 4 for simulation and inference. The maximum likelihood estimates and the standard errors are similar for Models 3 and 4 (Tables 2 and 3), so the difference between the two models is mainly in the log-likelihood height while shape and location are nearly identical. Since it is less costly in computer time to fit Model 4 to simulated data, we retained this model for inference by parametric bootstrapping. The gamma shape parameter α>0 is basically estimated from the numbers of recaptures by capture history within individual surveys, particularly their over-dispersion relative to the Poisson distribution. The estimate hits its upper boundary ( ). We infer from this that there is hardly any overdispersion in the capture intensity across individuals, and we neglect any such heterogeneity in the simulations. Note that the matching probability is much higher within than between surveys (see below). The absence of individual random effects in the capture intensity must be understood on this background. Some parameters are only weakly identified. The carrying capacity K hits its upper boundary ( ), and the shape parameter for the density dependence z is consequently also difficult to estimate. This is, however, no surprise. We use, in fact, only data from 10 years of photo surveys from 1984 to 2003 when the population appears to grow exponentially. That ˆK hits its upper boundary should, therefore, not be interpreted as a valid point estimate of carrying capacity. The year effect on capture intensity is assumed to be lognormally distributed, λ y = λ exp(σ u y ) with u y being standard normally distributed. Predicted values of the 10 random effects u y are in chronological order: 4.3, 3.0, 2.8, 2.9, 0.2, 0.4, 1.3, 0.7, 0.8, 0.3 relative to years. An initial capture is the first capture during a survey in a capture history. An initial capture might be followed by subsequent captures during the survey. The distribution of the number of subsequent captures by survey and capture history is also predicted through the predicted random year effects (including the 1984 data). This predicted distribution over 0, 1, 2, 3, 4, 5 subsequent captures, (4,906.4, 25.8, 2.0, 0.9, 0.6, 0.3) is compared to the observed distribution of subsequent captures (4,889, 40, 5, 1, 1, 0) by survey and capture history. Point estimates for the logistic intercept is 8.5, and for the linear effects of degree of marking at rostrum, mid-back, lower back, fluke zones are, respectively, 0.61, 0.76, 0.00, 0.55, while the effect of image quality for these areas are, respectively, 0.64, 0.44, 0.31, The log odds ratio for within survey matching is estimated as 9.3 relative to that between surveys. We regard the logistic parameter β as a nuisance parameter, and give only point estimates without standard errors or confidence bounds. Figure 1 presents the estimated matching probability as a function of the linear predictor and also shows a histogram of estimated individual matching probabilities. Marginal distributions of bootstrap estimates are presented as histograms in Figure 2 for a selection of parameters. Confidence curves are presented in Figure 3.

12 12 T. SCHWEDER ET AL. Figure 1. Logistic matching probability by estimated linear predictor (left), and histogram of match probabilities for individuals captured in one or more surveys (right). 6. DISCUSSION 6.1. HETEROGENEOUS MARKING AND VARIABILITY IN THE CAPTURE INTENSITY Data from the systematic photographic surveys of BCB bowheads from 1984 to 1998 were used to estimate population parameters of interest such as population size (Rugh Figure 2. Marginal histograms of 1000 simulated maximum likelihood estimates (Model 4) for initial abundance (1984) (top left), growth rate (top right), mortality intensity (bottom left), and median capture intensity (bottom right). True values assumed in the simulation are marked by vertical lines (true value for initial abundance (1984) = 3,952; for growth rate = 0.03; for mortality intensity = 0.017; for median capture intensity = ).

13 PHOT-SURVEYS OF NATURALLY MARKED WHALES 13 Figure 3. Confidence curves (Model 4) for abundance in 1984 (solid line), 2001 (dotted line), and 2007 (broken line) (top panel); yearly growth rate (middle left); yearly mortality intensity (middle right); median of hourly capture intensity (bottom left); and variation of hourly capture intensity (bottom right). The horizontal lines represent confidence The vertical lines indicate 95% confidence intervals. 1990; da Silva et al. 2000; Schweder 2003; da-silva, Gomes, and Stradioto 2007) and survival rates (Zeh et al. 2002). We followed Schweder (2003) in modeling the times of captures as a conditional Poisson process for each individual and allowed for random variation in capture probability as in Zeh et al. (2002), but unlike previous studies, we do not dichotomize between marked and unmarked individuals. This allows for false negatives (resightings of whales that are not recognized) in the matching process. As evident from Figure 1, for a vast majority of initial captures, the probability of a true recapture being recognized as a match is small. The estimated probability of successful matching is highly skewed, with only 0.4% of captures having a matching probability above 80%. The 95% of the captures with matching probability less than 0.1 might be regarded as unmarked. The 1.3% of the captures with matching probability above 0.5 are about evenly spread

14 14 T. SCHWEDER ET AL. over the matching probability interval Categorizing whales as simply marked or unmarked, as done in previous studies, therefore, loses information. The predicted values for the random year effects u y are unfortunately not in good agreement with the normal distribution. The tails are too heavy. A remedy could be to use a heavy tailed distribution, say a t-distribution, rather than a normal distribution for the random year effect on capture intensity. We doubt that this would make a substantial change to the estimates of abundance and mortality, and have not followed up on this. The predicted numbers of initial captures by year do, however, agree with the corresponding observed numbers (not shown). The predicted distribution of number of subsequent captures (see Section 5) is slightly shifted to the left relative to the observed distribution, but shows basically the same degree of overdispersion. This lack of fit might be due to the increased rate of subsequent captures being accounted for in the model by the within survey matching effect in the logistic regression. This might not be adequate, and the estimated effect of 9.3 is quite a bit more than expected. The numbers of subsequent captures are, however, only used to estimate the variability in the capture intensity across individuals within the respective survey. Since we found this variability to be negligible, the lack of fit is of little concern. However, eliminating the survey effort in 1984, which was estimated using the number of survey days, from consideration, the predicted values for the random year effects u y are in better agreement with the normal distribution (predicted values of these random effects are:, 0.9, 0.9, 1.1, 0.4, 0.2, 1.1, 0.1, 0.6, 0.4 relative to years). But by eliminating the survey effort in 1984, the MLE estimate of the yearly growth rate goes to zero, while estimates for the other parameters are nearly the same. This is due to the capture rate in 1984 being much smaller than in the other years, and eliminating this year from consideration, the capture rate shows a decreasing trend. Despite the equivocal status of the data from 1984, we decided to base our results on data that include that year. The main reason is that we are fairly confident with our estimate of total survey effort in COMPARISON WITH OTHER RESULTS FOR BOWHEAD WHALES Our point estimate of yearly growth rate is 3.2% and of abundance in 2001 is 8,250 (Table 2). These estimates are not significantly different from the most recent estimates for BCB bowhead whales made using ice-based visual survey data (George et al. 2004). Zeh and Punt (2005) estimated the annual growth rate of the population to be 3.4% and the abundance in 2001 to be 10,545, with respective 95% confidence intervals (1.7%, 5.0%) and (8,200, 13,500). Using a larger part of the same bowhead whale photography dataset (i.e., including data from summer surveys), Zeh et al. (2002) estimated the mortality to be 1.6% by a Bayesian Markov Chain Monte Carlo implementation of the Jolly Seber model. We estimate the mortality rate to be 1.7% per year, with 95% confidence interval (1.1%, 2.3%). Brandon and Wade (2006) showed that K could not be estimated with useful precision with a time series of relatively precise abundance estimates over a period of 24 years. When 150 years of catch data were also included, the carrying capacity of BCB bowhead

15 PHOT-SURVEYS OF NATURALLY MARKED WHALES 15 Table 4. For each marking zone: recaptured individuals with no worse quality of image at recapture distributed over change in degree of marking from capture to recapture n Rostrum Mid-back Lower back Fluke whales could be estimated. We had no hope of providing useful estimates of K and z from the photo survey data alone. The Pella Tomlinson model has rather been entertained to prepare for an integrated analysis where catch data, abundance data from visual and acoustic surveys, age data, and possibly other data, are brought together with the data from the photo surveys in a collective likelihood. It is also of some interest to note from Table 2 that the information wasted by including the two parameters for density dependence is rather negligible ASSUMPTIONS FOR CONFIDENCE INFERENCE A simulation experiment with 27 runs of 1000 replicates each was carried out to check whether bias is invariant across parameter values near the estimated parameter point, and whether acceleration in Efron s abc method can be neglected. The design is a full factorial in the three parameters ρ, μ, and N, which were increased by 10%, kept at estimated values, or decreased by 10%. The other parameters were kept at estimated values. Results are not shown in tables or figures. No systematic effects were found in linear analysis of bias. But there were some statistically significant quadratic effects of small magnitude for all parameters for bias. Acceleration parameters in the abc bootstrap method are estimated to be 0.1 for abundance and slightly smaller values for the recruitment and mortality parameters. All the estimates of acceleration of error were statistically insignificant. We found these estimates small enough to allow acceleration to be neglected when computing confidence curves MEASUREMENT ERRORS IN MARK READING The distribution of recaptures by change in degree of marks from capture to recapture, with quality of marking zone image at least no worse than in the recapture, was considered as a criterion for estimating measurement error in categorizing degree of marks (Table 4). Marks, as visible in aerial photographs, are regarded to be permanent in bowhead whales. They are acquired over the life of the individual, and degree of marking thus, should not decrease from capture to recapture. The nonnegligible frequency of individuals that are scored with less marking at recapture than capture (Table 4) does indicate that measurement errors occur in the scoring of marks. These measurement errors have not been accounted for in our model and analysis.

16 16 T. SCHWEDER ET AL FALSE POSITIVE MATCHES False negative matches (resightings that are unrecognized) are indeed possible in photo surveys, and we discussed methods for handling potential erroneous capture histories stemming from these false negatives. Most false negatives are due to poor image quality or whales being poorly marked. False positives are also conceivable, such as when a captured individual is erroneously matched with a previous capture. However, we believe that false positives are rare among these photographic data for two reasons: (1) The paired images are available for discussion and review because there are so few of them (unlike the search for matches through the whole dataset which is done by only one or two people). Furthermore, three people must agree that a match is confirmed before it is applied to the analysis. This procedure (Rugh et al. 1998) probably results in some real matches being rejected, but it effectively eliminates the probability of false positives. (2) Other data in the photographic database, such as whale length, are available to aid in the final decision about matching whale images. For instance, if one whale is 15.0 meters long, and a later potential match is 13.0 meters long, it would not be considered a match. The probability of making a false match rises with decrease in image quality and decrease in uniqueness of marks; therefore, such images are unlikely to be included in the sample size of most statistical analyses used to apply rates of reidentifying individual whales because most previous studies merely limited the sample set to adequate images of well-marked whales. The more stringent the protocol is in attempting to prevent false positives, the more false negatives there will be. If analytic machinery were also available for handling false positive matches, one might try to tune the matching algorithm, with respect to stringency, to optimize the information content in the data. We have not developed such methods and have only the following vague ideas to offer. False positives constitute additional recaptures in some capture histories. It is possible, but unlikely, that a false positive is also a false negative, i.e., that a capture or recapture should have been linked to another capture. Accounting for false positives is a delicate matter, but is not impossible provided the protocol is operational to the extent the matching process can be simulated. The problem at hand is actually closely related to the problem of duplicate matching in double platform line transect surveys of minke whales. Skaug et al. (2004) accounted for both false positives and false negatives in duplicate matching by way of simulation. To make automation and simulation feasible for the very extensive photo archive, an elaborate tree structure might be required in the database CONCLUDING REMARKS Previous statistical analyses of the database resulting from the photo surveys at Barrow demonstrated that these data are valuable for estimating abundance and demographic parameters for the population of bowhead whales in the Bering Chukchi Beaufort seas. This study lends additional support to this view. However, the existing data are not fully worked up yet since captures from 2003 are not matched with previous captures. There has been an additional photographic survey in 2004, but data from these have not been worked

17 PHOT-SURVEYS OF NATURALLY MARKED WHALES 17 up, and more aerial surveys of the spring migration of bowheads are proposed. Therefore, the ongoing effort for collecting and examining photographs will continue to bolster the analysis conducted here. An aboriginal subsistence harvest of BCB bowhead whales is managed by the International Whaling Commission using abundance estimates obtained from ice-based visual and acoustic surveys (e.g., George et al. 2004). These ice-based studies rely on adequate weather and ice conditions throughout a major portion of the bowhead spring migration near Barrow. A secure ice shelf is required from which to make observations across an open stretch of water where migrating whales are visible. There is concern that global warming will result in a decrease in the number of years with suitable ice conditions to conduct the surveys. Abundance estimation by photographic surveys might be an attractive alternative. Schweder and Sadykova (2009) demonstrate that the archive of previous captures becomes progressively more informative, and they provide simulations showing that continued photographic surveying might lead to abundance estimates with precision that competes favorably with even the best of surveys from the ice. In a simple sequential capture recapture experiment the information gain (the increase in inverse expected variance) per capture grows, in fact, faster than linear in previous number of unique captures. A more direct count method, like the current ice-based visual survey, has a constant information gain per unit field effort, and will, therefore, sooner or later be out-competed by a capture recapture survey, even though the latter may in some years have limited field effort due to weather or fiscal restrictions. Photographic surveys might not replace shore-based visual surveys altogether, but they do, as we demonstrated, provide valuable information on the status of the BCB bowhead population, and results from the photo surveys should preferably be integrated into the regular assessments of the population. ACKNOWLEDGEMENTS The National Marine Mammal Laboratory (NMML) and LGL Ltd. (LGL) are the lead organizations in conducting aerial photography of bowhead whales and maintaining photographic collections and metadata. Many people assisted in the acquisition of photographs and data that are used in this analysis. The authors thank Robyn Angliss, Lisa Baraff, Mary Nerini, Kim Shelden, and Dave Withrow of NMML, and John Richardson and Gary Miller of LGL. Photographic surveys were conducted under Scientific Research Permits 580 and (NMML) and 670 (LGL) issued by the U.S. National Marine Fisheries Service under the provisions of the U.S. Marine Mammal Protection Act and U.S. Endangered Species Act. The surveys were funded primarily by NOAA Fisheries Service (NMML); U.S. Minerals Management Service; and the North Slope Borough with minor contributions by many other groups. Modeling and analysis was carried out under Norwegian Research Council grant We benefited from discussions with Judith Zeh. The authors thank the editor and two referees for their helpful comments. [Received September Revised March Published Online January 2010.] REFERENCES Angliss, R. P., Rugh, D. J., Withrow, D. E., and Hobbs, R. C. (1995), Evaluations of Aerial Photogrammetric Length Measurements of the Bering Chukchi Beaufort Seas Stock of Bowhead Whales (Balaena mysticetus), Report of the International Whaling Commission, 45,

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