Atlantic bluefin tuna in the Gulf of Maine, II: precision of sampling designs in estimating seasonal abundance accounting for tuna behaviour

Abstract

A primary challenge of animal surveys is to understand how to reliably sample populations exhibiting strong spatial heterogeneity. Building upon recent findings from survey, tracking and tagging data, we investigate spatial sampling of a seasonally resident population of Atlantic bluefin tuna in the Gulf of Maine, Northwestern Atlantic Ocean. We incorporate empirical estimates to parameterize a stochastic population model and simulate measurement designs to examine survey efficiency and precision under variation in tuna behaviour. We compare results for random, systematic, stratified, adaptive and spotter-search survey designs, with spotter-search comprising irregular transects that target surfacing schools and known aggregation locations (i.e., areas of expected high population density) based on a priori knowledge. Results obtained show how survey precision is expected to vary on average with sampling effort, in agreement with general sampling theory and provide uncertainty ranges based on simulated variance in tuna behaviour. Simulation results indicate that spotter-search provides the highest level of precision, however, measurable bias in observer-school encounter rate contributes substantial uncertainty. Considering survey bias, precision, efficiency and anticipated operational costs, we propose that an adaptive-stratified sampling alone or a combination of adaptive-stratification and spotter-search (a mixed-layer design whereby a priori information on the location and size of school aggregations is provided by sequential spotter-search sampling) may provide the best approach for reducing uncertainty in seasonal abundance estimates.

Appendix

Survey precision—bias and uncertainty

The optimal choice of an estimator depends on evaluation of a trade-off in measurement bias and precision. Mean squared error (MSE) is used as a standard criterion and is expressed as,

$$\hbox{MSE}(\hat {E}(N))=\hat {E}\left[ {\hat {E}(N)-\hat{N}}\right]^2=\beta (\hat {E}(N))^2+\hbox{var}(\hat {E}(N))$$

(A1)

where ${\hat{E}(N)},$ is expected population abundance, β is bias and var is variance in abundance, respectively. Under the central limit theory, bias and variance both decrease as estimates approach the expected or ‘true’ values. However, the extent to which bias and variance trade-off as a function of sample size and/or effort (e.g., number of transects, number of observers) is a strictly empirical question. Contributions to survey bias from empirical data indicate that the bias in estimating aggregation strength of schools lies within 30.7–32.1%, the bias in estimating school size is 166–317%, bias in estimating number of schools is 143–315%, with bias in movement of 48–65% (horizontally) and 50% (vertically/depth) (Newlands et al. 2006). Net-bias for these contributions is therefore 234–458%. Note that this net-bias estimate does not include observer bias in encountering schools from survey transects estimated to be 1,280–3,270% (Newlands et al. 2006) because this contribution to bias is likely an over-estimate (i.e., poorly estimated) given that it was obtained by pooling three separate year estimates and forming the percentage required assuming that a value of CV = 1 is 100% whereby distribution mean is equal to standard deviation. This assumption was required due to a lack of reliable information available on the statistical distribution of observer-school encounter rate, especially a spotter-search scheme. What must be determined is the maximum value of CV most likely to occur, so that our estimates could be scaled to such a maximum whereby CV = CV_max as 100% could then be assumed, instead of CV = 1. Survey precision is expressed in the coefficient of variation (CV), as a ratio of sampling mean to variance.

Survey efficiency

We define the efficiency of survey sampling as the increase in survey precision achieved for a threshold number of observers, o _T , relative to only one observer. Efficiency ${(\Upsigma)}$ is expressed in terms of survey precision, as

$$ \Upsigma =\frac{\hbox{CV}(o_T)-\hbox{CV}(1)}{(o_{T}-1)} $$

(A2)

For the purposes of comparing the efficiency of results for the five simulated survey schemes, a threshold (o _T ) of 20 observers was chosen.

Schemes A, B: random and systematic transect

Random transect sampling involves initially selecting the position of a transect line randomly, with random sampling conducted along the transect line. Spacing between transects was initially determined by uniform random sampling within an acceptable range. All transects were then systematically separated by the sample separation distance. Sampling along the transect line in systematic sampling is conducted by initially determining the sampling rate along a transect line, with all sampling spaced equally at this interval for all transect lines (Krebs 1999; Buckland et al. 2001).

$$\hbox{CV}(\hat{\omega}_s)=\frac{1}{\hat{E}(\omega_s)}\sqrt{\frac{\hbox{var}(\hat{\omega}_s)}{L_s}\left({\frac{L_T-L_s}{L_T}}\right)} \ (\hbox{without replacement})$$

(A3)

$$\hbox{CV}(\hat{\omega}_s)=\frac{1}{\hat{E}(\omega_s)}\sqrt{\frac{\hbox{var}(\hat{\omega}_s)}{L_s}} \ (\hbox{with replacement})$$

(A4)

Scheme C: systematic stratified transect

For this survey scheme, the coefficient of variation was estimated as (Krebs 1999; Buckland et al. 2001),

$$ \hbox{CV}(\hat{\omega}_s)=\sum\limits_{h=1}^{h^s}{\left({\frac{1}{\hat{E}(\omega_s)_h}\sqrt{\frac{\hat{\omega}_h^2\hbox{var}(\hat{\omega}_s)}{L_{s,h}}\left({\frac{L_T-L_{s,h}}{L_T}}\right)}}\right)} $$

(A5)

for h  =  (1, ... ,h ^s) sampling strata. In Eq. A5, ${\omega_h^2}$ is the proportion of school encounters (N _h /N) within each stratum, termed the sampling fraction for stratum h. Strata should be chosen so that sampling fraction is homogeneous, whereby greater precision is obtained when total variance across all strata h _s depends only on the variances within stratum.

Scheme D: adaptive stratified

Adaptive stratified sampling involves initial probability selection of grid units (termed primary units), whereby additional grid units are added to a sampling in the neighbourhood of any selected unit for which the observed value of a variable of interest satisfies a specified condition, termed secondary units. The purpose of adaptive sampling is to take advantage of population characteristics (such as aggregation) in their distribution to obtain a more precise estimate of population parameters (i.e., school encounter, density, abundance) with a given amount of survey effort. In the simulated adaptive sampling scheme, π _k denotes the probability than one or more primary units are included within a transect sample. This probability, termed an intersection probability, is given by,

$$ \pi_k={1-\left(\begin{array}{c} {N-x_k} \\ n\\ \end{array}\right)} \left/{\left(\begin{array}{c} N \\ n\\ \end{array}\right)}\right. ={1-q_k}\left/{\left(\begin{array}{c} N\\ n\\ \end{array}\right)}\right. $$

(A6)

where x _k , is the number of primary units (i.e., first schools sighted in a survey), N is the total number of grid units, and n is the total number of grid units sampled. Based on primary number of schools detected by observers, secondary detections are added to the survey sample with secondary inclusion probability, π _kj . The collection of all primary units and all secondary units are termed primary and secondary networks, respectively, with index k. The total number of networks is K. Considering only 2 layers, then K = 2. The secondary intersection probabilities are determined as

$$\pi_{jk}={1-\left[{\left({{\begin{array}{c} {N-x_k}\\ n\\\end{array}}}\right)+\left({{\begin{array}{c}{N-x_j}\\n\\\end{array}}}\right)-\left({{\begin{array}{c}{N-x_k -x_j+x_{kj}}\\n\\\end{array}}}\right)}\right]}\left/{\left({{\begin{array}{c} N\\ n\\\end{array}}}\right)}\right. ,$$

(A7)

re-expressed as

$$ \pi_{jk} ={1-q_k -q_j -\left({{\begin{array}{c}{N-x_k -x_j +x_{kj}}\\n\\\end{array}}}\right)}\left/{\left({{\begin{array}{c} N\\ n\\\end{array}}}\right)}\right. ,$$

(A8)

where x _kj is the number of primary units that intersect the primary and secondary networks k and j, respectively. The sampling fractions, (q _k , q _j ) are defined as (1−π _k ) and (1−π _j ). An indicator variable z _k was defined to equal unity if one or more primary units that intersect network k are included in the initial sample, and zero otherwise. The mean estimator of abundance, ${\hat{t}_3},$ for a general variable of interest y _k is (Thompson 1991a, b),

$$ \hat{t}_3=\frac{1}{MN}\left({\sum\limits_{k=1}^K{\frac{y_k z_k}{\pi_k}}}\right) $$

(A9)

The ${\hat {t}_3}$ estimator is a summation over all distinct networks (K  =  2) in the sample that intersect one or more primary units of an initial sample. In Eq. A9, M and N denote the total number of schools detected in the primary and secondary network, respectively. Statistics of the defined indicator variable are related to the intersection probabilities. The mean of the indicator variable, z _k , is ${\hat{E}(z_k)=\pi_k}$ and ${\hat{E}(z_k z_j)=\pi_{kj}},$ with variance, ${\hat{\sigma}^2(z_k)=\pi_k\left({1-\pi_k}\right)}$ and covariance, ${\hat{\sigma}^2(z_k ,z_j)=\pi_{kj}-\pi_k\pi_j},$ where it is assumed i ≠ j. Here, ${\hat{\pi}_{kj}}$ is the joint intersection probability between the primary and secondary grid networks. With the convention that ${\hat{\pi}_{kk}=\hat{\pi}_k},$ the variance in the ${\hat {t}_3}$ estimator is

$$ \hbox{var}({\hat {t}_3})=\frac{1}{({MN})^2}\sum\limits_{k=1}^K{\sum\limits_{j=1}^K {y_k y_j\left( {\frac{\pi_{kj}}{\pi_k\pi_j}-1}\right)}} $$

(A10)

An unbiased estimator of the variance for adaptive sampling involving an environmental indicator variable, z _k , is re-expressed as (Thompson 1991a, b),

$$ \hbox{var}({\hat {t}_3})=\frac{1}{({MN})^2}\sum\limits_{k=1}^K{\sum\limits_{j=1}^K{y_k y_j z_k z_j \left({\frac{\pi_{kj}}{\pi_k\pi_j}-1}\right)}} $$

(A11)

Scheme E: spotter-search

For the irregular behaviour and observed variability in spotter-searching and spatial sampling, the spotter-search scheme was formulated, not as an objective model, but using statistical bootstrapping of actual recorded spotter flight paths.