Buckland et al.: Estimating abundance of tuna-associated dolphin stocks in the eastern tropical Pacific 



efficiency of search of the tuna vessels. This parameter, 

 the effective search width, is estimated using line- 

 transect theory. It may be interpreted as twice the 

 distance at which the number of undetected dolphin 

 schools closer to the vessel is equal to the number of 

 detected schools further from the vessel, and is there- 

 fore the effective width of the strip of ocean searched 

 by the vessel. As efficiency of the fleet to detect dolphin 

 schools increases (e.g., through the use of helicopters, 

 high-resolution radar, etc.), the effective search width 

 increases, and bias in abundance estimates should re- 

 main unaffected. 



We adopt a strategy of reducing bias as much as 

 possible, so that the effect of any trend in bias over 

 time on estimated trends in abundance is minimized. 

 To estimate the different components of the estimator 

 of Equation (3), separate stratification schemes are ap- 

 plied for encounter rate, effective search width, and 

 school size. In stratifying for a given component, our 

 aim is to define strata such that each stratum is 

 relatively homogeneous with respect to that compo- 

 nent, so that non-random search effort and non-random 

 distribution of schools generate only small bias in any 

 given stratum. Crude encounter rates, average school 

 sizes, and average detection distances are estimated 

 by 1° square. Where data are insufficient, the crude 

 estimates are smoothed, and the same smoothing pro- 

 cedure interpolates for squares in which there was no 

 tuna vessel effort. These estimates are used to allocate 

 1 ° squares to strata, yielding the separate stratifica- 

 tions for encounter rate, school size, and effective 

 search width, respectively. Full details are given by 

 Anganuzzi and Buckland (1989). 



Thus the problem of abundance estimation has been 

 reduced to three simpler problems: For a random point 

 in the stock area, the expectations of encounter rate, 

 school size, and effective search width are estimated, 

 and the three estimates are multiplied together to ob- 

 tain the final abundance estimate. Lack of indepen- 

 dence between the three estimates does not bias the 

 overall estimate, and independence is not assumed 

 when estimating variance. A nonparametric bootstrap 

 technique is used to obtain variances. The resampling 

 unit in the bootstrap is the individual cruise, and for 

 each bootstrap replicate the full estimation procedure 

 is applied, thus generating bootstrap estimates of abun- 

 dance. The sample variance of these estimates yields 

 the required variance estimates, and confidence inter- 

 vals are obtained by the percentile method. (See Buck- 

 land and Anganuzzi 1988a, for details.) 



Bias arising from rounding errors in the recorded 

 sighting distances r and angles 9 is reduced by smear- 

 ing the data, using the method favored by Buckland 

 and Anganuzzi (1988b). The recorded location of 

 each school relative to the tuna vessel at the time of 



detection is defined by r and 9, and that location is 

 "smeared" over the sector defined by r • (1 ± d ) and 

 9 ± ^12. to allow for inaccuracy in the recorded values. 

 The smearing parameters d and I are estimated from 

 the data. When a small sighting angle is rounded to 

 zero, the calculated perpendicular distance is zero, 

 giving a spurious spike in the perpendicular distance 

 distribution at zero distance. Smearing yields more 

 robust estimation by removing or reducing this spike. 



Here we take the estimates of Anganuzzi and Buck- 

 land (1989) and of Anganuzzi et al. (1991) and attempt 

 to estimate the underlying trends in dolphin abundance 

 by smoothing them. Various smoothing methods such 

 as moving averages, running medians, and polynomial 

 regression were investigated (Smith 1988). The chosen 

 method was a compound running median known as 

 "4253H, twice" (Velleman and Hoaglin 1981), which 

 is constructed as follows. 



Suppose that {X{t )}, ^ = 1, . . . , A'^, is a time-series of 

 length A^, and let {5,(0} be a smoothed version of it, 

 found by calculating an i -period running median. We 

 can construct compound smoothing methods such as 

 {Sijit)}, which is simply {Sj{Si{t))}. Thus, a 4253 run- 

 ning median method smooths a time-series using a 

 4-period running median, which is in turn smoothed by 

 a 2-period running median, smoothed again by a 

 5-period running median, and then by a 3-period 

 running median (i.e., {54253(0} = {5'3(S5(S2(S4(0)))})- 

 Near the endpoints, where there are not enough values 

 surrounding a point to be smoothed using the spe- 

 cified running median, a shorter-period running median 

 may be used. The endpoints of the resultant time-series 

 are calculated by estimating X(0) and X(N + l), the 

 "observed" values at t = and t=N + l, and then 

 calculating 



54253(1) = median {1(0), X{1), 54,53(2)} and 



54253(iV) = median {S4253(A^-1), XiN), X{N + 1)}. 



X(0) is found by extrapolating from the straight line 

 which passes through the smoothed values att=2 and 

 i = 3, i.e., 1(0) = 3 -54953(2)- 2 -54953(3); similarly, 



X{N + 1) = 3- 54953(iV - 1) - 2 - 54253(^ - 2). 



The H in "4253H, twice" denotes a linear smoothing 

 method commonly used with running medians, which 

 is known as Banning. It is a 3-period weighted mov- 

 ing average iort=2,...,N-l, with weights {0.25, 0.5, 

 0.25}. The endpoints remain unchanged. 



The pattern of the time-series may be recovered by 

 calculating the residuals of the series (i.e., the differ- 

 ences between the smoothed and unsmoothed esti- 

 mates), smoothing the residual series using the same 

 method as for the time-series, and then adding the 

 smoothed values of the residuals to the smoothed 



