PENNINGTON: TECHNIQUES FOR ESTIMATING ABUNDANCE 



One method to construct an abundance index 

 based on the entire survey series is briefly as follows 

 (more details can be found in Pennington (1985)). 



Suppose the population (or z t ) can be represented 

 by the autoregressive integrated moving average 

 process (Box and Jenkins 1976, Chap. 4) 



0(5) z t = 0(5) a t . 



where the a t 's are independently identically distrib- 

 uted {iid) and normally distributed (N) with mean 

 zero and variance o 2 [iid N(0, o 2 )]. If y t = z t + e t , 

 and the e t 's are assumed iid N(0, o 2 ), then y t will 

 follow the model 



0(5) y t = r,(B) c t , 



(3) 



Suppose the factors causing the change in popula- 

 tion from year t - 1 to year t (such as recruitment, 

 fishing mortality, natural mortality, and migrations) 

 produce a t 's which are approximately iid N{0, o 2 ). 

 If the measurement errors are multiplicative, then 



In y t = In z t + e t . 



(8) 



Assuming the e/s are iid N(0, o 2 ) and independent 

 of the a/s, then it follows as above that y t can be 

 represented by the model 



(1 - 5) In y t = (1 - 05) c t . 



(9) 



where the c t 's are iid N(0, of) 

 For model (9) [generated by Equations (7) and (8)] 



where the c t 's are iid N(0, o 2 ). Now if model (3) 



and the ratio of/of are known, then the maximum and 



likelihood estimate of z t is given by 



= o*M 



2/„2 



(10) 



(1 - 0) 2 = olio 



&t = Vt ~ -z(Ct - nj c t+1 



- TIo C 



2 W + 2 ~ ) • • • » 



n T _ t c T ), (4) 



where T denotes the last year of the series, the c t 's 

 are the estimated residuals generated by model (3), 

 and the n values are calculated using the identity 



0(5) = (1 - n x B - n 2 5 2 - . . .) r]{B). (5) 



The variance of z t is given approximately by 



varfo) = o 



^ „2 



1 - (n§ + 4 



2 \ °e : 



(6) 



where rc = 1. 



The model for y t [Equation (3)] is usually ob- 

 tained in practice by fitting a model to the observed 

 series using procedures described in Box and 

 Jenkins (1976). If catchability is constant over time, 

 the within survey sampling variance provides an 

 estimate of o e z . But if catchability varies, another 

 approach is necessary. 



Toward this end, consider the expression 



"t-i 



or 



(1 - 5) In z t = a t . 



(7) 



Therefore, assuming the above approximations to 

 the population dynamics, fitting model (9) to the 

 observed survey series provides an estimate, 0, of 

 o^lol and an estimate of o 2 . The it-weights for the 

 model are from Equation (5) given by 



= (1 - 0) 0< 



i > 1. 



(11) 



It may be noted that if model (9) is valid and catch- 

 ability is constant over time then the estimate of o 2 

 given by d 2 [from Equation (10)] would approx- 

 imately equal the estimate of o 2 based on the within 

 survey sampling variance. 



AN APPLICATION 



The Northeast Fisheries Center conducts an 

 extensive groundfish trawl survey as part of its 

 MARMAP program two times a year: in the fall 

 since 1963 and in the spring since 1968 (Grosslein 

 1969). The survey region is divided into sampling 

 strata based on geographic boundaries and depth 

 contours (Fig. 2). For each survey, trawl stations 

 are chosen randomly within each stratum. One of 

 the objectives of the surveys is to provide indices 

 of abundance for the many species of commercial 

 value in the region. 



Yellowtail flounder is an important New England 

 fishery resource whose population has fluctuated 

 considerably over the survey period (Clark et al. 

 1984). Commercial catch statistics exist for yellow- 

 tail flounder, but age data suitable for a VPA (Vir- 

 tual Population Analysis) are unavailable. Major 



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