HOLT ET AL.: MONITORING DOLPHIN ABUNDANCE 



estimate of the probability density function of 

 perpendicular distances extrapolated to the 

 trackline. First, 



R 



i?2 ("' ~"^^ 



Var(n) 



i=l 



R 



where R is the number of replicate lines of equal 

 length (/). For R of moderate size, R =(R - 1). 

 Thus 



R 



Var in) - 2 ^"' " "^^ " ^^^^• 



1=1 



This is because Var (n) is the sum of the 

 variances of i? independent values (n,, i = 1, 2, ..., 

 R) each having the same expected variance. But 

 R = n/Eirii), the total number of sightings 

 divided by the expected number of sightings for a 

 line of length /. Thus, R = Oin), and 



Var in) 0(n) 



Oa/n). 



(6) 



n' 



Second, fiO) was estimated using a Fourier series 

 (FS) model (Burnham et al. 1980); therefore, 



Var[/(0)] = 2 2 Cov (d,,d^) 



^ = 1, 2, 3, ..., and k >j > 1 

 where the a's are the coefficients in the series 



ak 



nw 



n 



V^ jk-nx, 



cos 



1=1 



w 



= 0(1) 



with jc, equal the perpendicular distance to the ith 

 sighting and w equal the truncation point for the 

 perpendicular distance. Therefore, we only need 

 to know the dependence of cov(a,, Uk ) on n. If n is 

 much larger than one, in - 1) = n and 



Gov (dj, dk ) = 



n 



(dk+j + dk-j) - dj dk 



= oain). 



Since /"(O) estimates a quantity which is constant 

 with respect to n, 



Var[/(0)] 

 [/(0)]2 



Oilln). 



(7) 



Combining Equations (6) and (7) with Equation 

 (5), [CV (Z))]2 = Oain). This confirms discussions 

 presented by Burnham et al. (1980). 



In addition to investigating the theoretical de- 

 pendence of [CV (D )]2 on n, we tested its empiri- 

 cal dependence on n using the research vessel 

 data which included 479 days of survey effort. 

 Data were truncated at 3.70 km perpendicular 

 distance from the ship. Paired days of shipboard 

 searching effort were randomly selected using a 

 uniform random number generator until the 

 number of associated sightings (n) equaled or ex- 

 ceeded a previously selected sample size. Sample 

 sizes selected were 20, 30, 40, 50, 60, 80, 100, 200, 

 500, and 1,000. The resultant perpendicular dis- 

 tance distributions were smeared (a data smooth- 

 ing technique described by Butterworth 1982, 

 Hammond 1984, and Holt fn.8), and density, vari- 

 ance, and coefficient of variation estimates were 

 calculated for each data set. The simulation was 

 completed three times for each value of n . 



The relationship between CV (D) and l/Vn 

 (Fig. 3) was linear (/^lack-of-fit = 0.83; P = 0.59) 

 with intercept not significantly greater than zero 

 (t = 1.56; P > 0.10). This confirms the analytical 

 result above, that CV (D) = 0(1/Vn ); however, as 

 n increased, the probability of randomly selecting 

 data from each of the 240 pairs of days (479 sur- 

 vey days) multiple times increased which may 



7 r 



2 06 



z 



2 0.5 



= 04 



> 



Z 03 



UJ 



o 



iZ 2 

 u. 



UJ 



O 



O 0,1 



0.0 



-1—1 L_i I ' I i I I 



0,00 0.05 



0.10 15 



1/Vn 



0.20 



0,25 



Figure 3. — Comparison of number of dolphin sightings (l/Vn) 

 and precision of the population estimate (CV(D)). 



441 



