FISHERY BULLETIN: VOL. 81, NO. 2 



Analysis Methods: 

 Estimation of Abundance 



In fitting the probability density functions to the 

 census data, the unit used was the estimate of the 

 proportion of the population passing during a 24-h 

 day. The number passing on day ;' was estimated 

 as 



ftj= (SENAy)  24, (4) 



where E[n\ is the expected value of n, i.e., the es- 

 timate of the number per group, corrected for bias as 

 in Equation (1). The relative proportion passing on 

 day; was estimated as 



Pj = nj/JMj. 



(5) 



Model parameters were first estimated for each 

 year using all data points regardless of recorded 

 visibility conditions. Data were fit by the two- 

 parameter gamma model 



fU\**P) = 



a-r(/3) 



Ufa)?-* exp{-;/a} (6) 



for each migration separately. The parameters of the 

 gamma distribution, their variances and covariance, 

 were estimated by the method of maximum like- 

 lihood (Chapman 1956; Greenwood and Durand 

 1960). Equality of parameters between years was 

 tested by the.F statistic (Chapman 4 ), 



F = 



£ (jc - x) 2 /n - 1 

 £ var (x)/n 



(7) 



for x = a, [i. 



The distribution ofp ; for each year was then used to 

 determine the effect of visibility conditions on census 

 results. An average visibility condition was calcu- 

 lated for each day from all of the recorded codes (Ta- 

 ble 1). The difference (residual) between the 

 observed and predicted relative proportions for each 

 day was also calculated. An ANOVA was performed 

 on the residuals with visibility categories as groups, 

 along with multiple range tests (Duncan's, Student- 

 Newman-Kuels, Scheffe's). These results were used, 

 along with an examination of the mean squared errors 

 for each category, to set a critical level of visibility 

 conditions beyond which there was significant inter- 

 ference with accurate censusing. The data were then 



4 D. G. Chapman, Director, Center for Quantitative Science, College 

 of Ocean Fishery Sciences, University of Washington, Seattle, WA 

 98195, pers. commun. March 1980. 



refit by the gamma distribution using only days with 

 visibility codes less than the critical value as points. 

 The new set of daily predictors (p' ; ) from the fitted 

 gamma model were used in the further estimation 

 procedures. 



Then, as an alternate to Equation (2), the abun- 

 dance for day; was 



n, 



= f [(LE[n])A,] 24 

 WW) 



vis < critical value (8a) 

 vis > critical value. (8b) 



That is, for days with visibility conditions less than 

 or equal to some critical level (with levels defined as 

 in Table 1) the average hourly sighting rate, correct- 

 ed for counting bias, multiplied by 24 h, was used as 

 the estimate of the total number of whales passing. 

 For days with visibility conditions worse than some 

 critical value, the estimate of the number passing 

 came from the expected proportion for the day (from 

 the gamma distribution model of migratory timing for 

 that year, p'j) multiplied by the sum of the daily es- 

 timates from the first fitting of the gamma model. 



For estimating the "tails" of the migration, a slight 

 modification of the method of Mundy (1979) was 

 used. This method was developed to predict total run 

 size for salmon from intermediate results of counts, 

 given that migratory timing can be modeled. The to- 

 tal "run" Nj was predicted by minimizing the least 

 squares error function 



/ En, . , 



(9) 



which was solved for Nj {N estimated by data cumu- 

 lative to day;') by 



N,=?(2>i ; ) 2 /2^, 



(10) 



Here Mundy uses 6 ] as the cumulative proportion ex- 

 pected to have passed by day;, and we define 0, as 

 that quantity less the predicted proportion missed 

 before the first day of each census. 



The final form of the abundance estimate for each 

 year k was then, 



A^ = {Z(Zn//(En,)-0,)/7(*). (ID 



The variance for Equation (11) was estimated in 

 two ways. The first, S5, outlined in Appendix 2, was 

 derived from the component variances of the pa- 

 rameters used in the model, employing the Delta 

 Method (Seber 1973). In the second method the data 

 were subsampled in five 2-h samples/d. The five 



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