Sigler: Estimation of abundance of Anop/opoma fimbria off Alaska 



593 



Table 1 



Data sets used in the age-structured model of Alaskan sablefish. Length data are specified by sex. 



Data component 



Data aggregation 



Years of data 



Longline survey relative abundance 

 Longline survey male lengths (cm FL) 

 Longline survey female lengths (cm FL) 

 Longline survey age 

 Fisherv total catch 



40-41. 42-43 62-63, 64-69. 70-74, ..., 95-99 



40-41, 42-43 68-69, 70-74, 75-79, ..., 95-99 



2, 3, 4, 5, 6, 7, 8, 9-10, 11-15, 16+ years 



1979-95 



1979-95 



1979-95 



1983, 87, 89, 91, 93 



1979-95 



The modeled discrete fishery misrepresents the 

 duration of the earher year-round fishery, but tests 

 with simulated data comparing a continuous fishery 

 showed little difference in estimated abundance. The 

 assumption of a discrete fishery simplified catch ac- 

 counting and reduced model convergence time, which 

 was important for the Monte Carlo simulations. This 

 formulation limits the maximum exploitation rate 

 for an age class to equal the selectivity for that age. 

 This maximal rate would be achieved if the maxi- 

 mally selected age were completely harvested, with 

 the exploitation rate equal to 1.0. This potential prob- 

 lem is not a real one because exploitation rates never 

 approach such high levels for Alaskan sablefish. 



Selectivity was described by the "exponential-lo- 

 gistic" function (Thompson, 1994), 



U-yJ 



i-y 



PriA,,„-a> \ 



[1 + e 



litA,,"- 



The "exponential-logistic" function is flexible, allow- 

 ing both asymptotic selectivity when selectivity in- 

 creases with age to an asymptote, and dome-shaped 

 selectivity when selectivity increases with age to a 

 maximum, then decreases for older fish. The expo- 

 nential-logistic function automatically scales maxi- 

 mum selectivity to 1.0 and reduces to asymptotic 

 selectivity as the parameter gamma (7) approaches 

 zero. When y= 0, the parameter A^g is the age where 

 500r of the population is vulnerable and /J is the slope 

 of the function at Ag^. When 7 > 0, then A^g and P 

 lose their biological meaning, because A^q no longer 

 represents the age at 50'^^ vulnerability. Selectivity 

 is assumed equal for the fishery and survey. Both 

 the fishery and survey mostly use longline gear and 

 cover the same area. Their similar length frequencies 

 support this assumption (Fig. 3). The fisheiy length 

 data were not incorporated into the model because of 

 this similarity and because few years of fishery length 

 data were available. 



Age data were aggregated over adjacent ages 

 (Table Das suggested by Derisoetal.( 1989) because 



sablefish are difficult to age, especially those older 

 than eight years (Table 6 in Kimura and Lyons, 1991). 

 Ages greater than eight years were not pooled into a 

 single class because females continue to grow, though 

 slowly, after eight years. This growth information was 

 needed in the model to convert ages to lengths. 

 Length data also were aggregated (Table 1 ). Another 

 way to deal with ageing error is to include a matrix of 

 misageing probabilities in the age-structured model 

 (Kimura, 1990), but this approach was not used be- 

 cause the misageing probabilities were unknown. 



Estimated data values were computed from the 

 parameter estimates. The estimated abundance in- 

 dex is I^ = qN[. ,where q is survey catchability. Quan- 

 tities estimated from this model and used in the 

 model fitting algorithm are denoted with "hats." The 

 estimated age compositions are 



N 



Pva 



/ , V 



An age-length transition matrix, L = (/^^p, also was 

 calculated from the sui-vey age data, where /^^, is the 

 probability that a sampled fish of age a will be of 

 length /. The age-length transition matrix is assumed 

 constant over time. Probabilities were computed by 

 region and year, then pooled as an average weighted 

 by sample size. Estimated age compositions were 

 converted to estimated length compositions with the 

 age-length transition matrix. 



Pv/ 



^P.Jal 



The parameters were estimated by maximum like- 

 lihood by assuming multinomial errors for age and 

 length data and log-normal errors for catch data 

 (Fournier and Archibald, 1982) and by using the 

 quasi-Newton algorithm implemented in Microsoft 

 Excel. Assuming that the effective sample sizes are 

 the same for age and length samples and that the 



