Analytical methods generally follow those described in our previous analysis of fin whale abundance trends [14]. A brief description is provided here.

Process and Observation Models.

Models were developed separately for B. bairdii, Z. cavirostris, a single Mesoplodon species group, and a group of unidentified ziphiids (which were either Z. cavirostris or Mesoplodon), although sighting-distance data were pooled across these groups for purposes of estimating parameters of the detection function. Recognizing that B. bairdii are more easily detectable than other ziphiids (they are larger, occur in larger groups, and have more conspicuous blows and surface behavior), the detection model included covariates for inter-species differences (see below).

Following [14], the model for each species group is partitioned into process and observation components. The process model describes how animal density (D) changes through time, so that abundance at time t, N t = D t * A, where A is the size of the study area. The most general model we considered describes variation in animal density simply as a function of a single temporal trend parameter (β 1 ) and a stochastic error component (random variable, γ t ), for each year (t). Small sample sizes precluded more complex (e.g., geographically stratified) models. If the population is changing exponentially, the full density model is: (1) ∼Normal(0, σ).

The observation model links the state process to the observed data. Following line-transect sampling theory [19], and treating the observed counts of groups each year as a Poisson random variable [14]: (2)where n t is number of groups detected; s is a single mean group size estimate for the species (there was no evidence of annual variation) with overdispersed Poisson variance (see [14]); f t (0) is the value at distance y = 0 of f t (y), which is the pdf of the detection probability function g t (y), with g t (0) being the detection probability on the transect line; and L t is the on-effort transect length (km), considered to be measured without error (Table 1). If variance in the observed counts is over-dispersed (i.e., extra-Poisson), this should be handled implicitly by the process error term in equation 1. This can be seen by substituting the expression for D t (eqn 1) into equation 2 and re-arranging slightly so that the error term, γ t , moves outside of the density term: Thus, we may think of γ t as the sum of γ t ,p +γ t ,s+ , where subscripts p and s+ refer to process error and extra-Poisson sampling error, respectively. Estimates of γ t ,s+ in individual years from bootstrapping methods (e.g., [18]) could potentially be used to obtain more explicit estimates of process variance; this would be useful for projecting future abundance estimates with greater precision.

A more intuitive expression of equation 2 is: (3)where w equals the data truncation distance (4 km in our case) and q t is the average detection probability of a group within the surveyed area 2L t w. Equation 3 thus indicates that the expected number of groups detected equals the group density, multiplied by the area surveyed and the average detection probability within the area surveyed, defined as q t = g t (0)/f t (0)·1/w. In other words, q t is the “effective strip half-width” [1/f t (0)] divided by the total distance from the vessel within which searching takes place and corrected for imperfect detection on the trackline. The effective strip half-width is a mathematical re-interpretation of the distance function g t (y) into a single theoretical distance from the transect line within which groups have a detection probability of 1 and beyond which the probability is zero.

Detection probability decreases as Beaufort sea state increases. Thus the estimate of q t in equation 3 is: where L b,t is the amount of survey effort in each of five Beaufort categories (b = 1 (for classes 0 and 1), 2, …5) in year t, and q b = g b (0)/f b (0)·1/w. Note, the estimate for is calculated from the effort-weighted mean of the ratio [g b,t (0)/f b,t (0)], not the ratio of the means . Based on previous analyses in our case study system [18] we assume a half-normal detection function for g b (y): where h denotes half-normal parameters and the proportionality sign is used since g(0) may be less than 1. We estimated the scale parameter σ h,b and hence f b (0) as a function of covariates [20], assuming the following model: (4)where β h 0 is the intercept; and β h 1 and β h 2 are the coefficients for Beaufort sea state and the log of mean group size for the species, respectively (we use log of group size following convention of earlier SWFSC cetacean abundance analyses [14], [18]). The covariate model is based on the one used by Barlow and Forney [18] for beaked whales, the main difference being that we did not include a categorical variable for the ship on which observations occurred. Preliminary analyses did not reveal this variable to have much importance on parameter estimates, while it complicated the weighted-mean estimation of . Species group (B. bairdii vs. other/smaller species) was initially considered as a covariate as well (and was included in a Sensitivity analysis – see Results), but the sample size for B. bairdii was small (Table 1); preliminary analyses suggested that group size was a more useful variable overall and sufficiently acted as a proxy for B. bairdii since they usually occur in larger groups. As sample sizes for B. bairdii increase with future surveys, a separate variable for them should be included. The parameters for equation 4 were estimated from data for individual detections: where i denotes each observed group (all species detections pooled).