Perkins and Edwards: Capture rale as a function of scfiool size for Stenel/a attenuata 



547 



ments and did not attempt to correct for the possi- 

 bility of size estimation biases in either data set (see 

 "Discussion" section) or use a deconvolution kernel 

 to account for estimation variance. 



Our estimates of wJs) were based on modelling 

 the inherent selection bias in the research vessel 

 sighting data. We used a bivariate hazard-rate de- 

 tection function in a size-dependent line transect 

 analysis of the perpendicular sighting distances and 

 sizes of the observed schools (Drummer and 

 McDonald, 1987; Palka, 1993). Perpendicular dis- 

 tances were binned to reduce the effect of rounding in 

 the data. School sizes were not binned because we did 

 not use a parametric model for their distribution. 



We define the average capture frequency for a 

 school of size s as 





(n 



■hool: 



Setting the observed counts iig^i,^,^i^ and n^^^^ equal to 

 their expectation gives 



2Li 



•■ff 



'^schools ~ ^['hchonlsl  



^sets ~ '^['^setsi ~ t trips'^ sets i 



N.. 



■hoob 



and 



where L 

 A 

 f 



' trips 



the total distance searched by the re- 

 search vessels; 



the total area within the stock bound- 

 aries; and 



the fraction of tuna vessel trips that 

 carried an observer. 



Using the relationship between /rts) and if'^is), and 

 these moment equations for n^^^^^,^, and n,^,,,., we esti- 

 mated the capture frequency for a school of size s as 





Note that the factor w ,,- cancels out and only w As) 



eft •' eft 



remains. 



Because there were so few schools smaller than 

 100 animals set on by tuna vessels, and so few schools 

 larger than 1000 animals sighted from research ves- 

 sels, we restricted our analysis to schools from 100 

 to 1000 animals, and computed estimates of capture 

 frequency at intervals of 100 animals. 



_ Stratification and pooling There were many school 

 size data from tuna vessel sets (n , =3454), however, 



sets ' 



there were far fewer research vessel sighting data 



= 499) and these had an uneven spatial distri- 

 bution. We decided that research vessel sightings were 

 too sparse to stratify our estimates of ;r*(s) and wJs) 

 geographically. We did make stratified estimates ofp(s), 

 but this had very little effect in absolute terms on the 

 estimates of capture frequency, and so we present only 

 pooled estimates for simplicity. 



Similarly, we did not stratify by year in any of our 

 estimates. Set data, trip sampling fractions, sight- 

 ing data, and search effort were combined to make a 

 single estimate of the average N^^^^ and N^^i^^^g^ over 

 all years. 



Extrapolation to the international fleet and to annual 

 estimates Data from individual sets came only from 



U.S. tuna vessels, so that estimating capture fre- 

 quency due to the entire fleet required extrapolation. 

 We made the assumption that a captain's preference 

 for dolphin school sizes upon which to make sets did 

 not vary with the vessel's country of origin and thus 

 extrapolated our estimate of p(s) to the entire fleet. 

 Our estimates of total numbers of target sets were 

 based on separate observed counts and sampling frac- 

 tions for the U.S. and the international fleets, 



sets sets sets 



"sets 

 /■It/Si 

 f trips 



Jlntl) 



Itr 



ilnth 

 ips 



This expression was substituted in for '",(,(j //^rip.^i 



in our estimates of capture frequency for the com- 

 bined fleet. For non-U. S. vessels, we had only the 

 total number of target sets observed, and we were 

 unable to estimate the variance in N'^""\ Thus, our 



sets _ 



estimates for the combined fleet do not include esti- 

 mates of precision. 



Our estimates of capture frequency are strictly 

 valid only for the study period. However, if we as- 

 sume that the same patterns in school sizes and 

 captain's preference for school sizes hold for the en- 

 tire year, then the annual capture frequency can be 

 estimated by using the corresponding annual set 

 counts for U.S. and non-U. S. vessels. Because we had 

 only the total number of target sets observed during 

 times other than the study period, our estimates of 

 annual capture frequency do not include estimates 

 of standard error. 



Independence of observations 



Because of the geographically correlated nature of 

 consecutive research vessel sightings or tuna vessel 

 sets, successive school size observations from a single 

 vessel may not have been independent. This is par- 

 ticularly a concern for the set data, because of the 

 possibility of repeated sets on the same school (see 



