760 



Fishery Bulletin 90(4|. 1992 



Relative abundance of armorhead was therefore based 

 solely on Japanese data and was calculated as the 

 reported monthly catch divided by the fishing effort 

 (in hr). Annual mean catch-per-hour (Ut) and its vari- 

 ance were calculated from the unweighted monthly 

 means. FI could not be calculated during this period 

 due to insufficient data. 



Biomass estimation 



Armorhead abundance could be estimated in absolute 

 terms as biomass only in Period 2. In the other periods, 

 abundance was estimable in relative terms as CPUE. 

 To allow estimation of biomass from CPUE in Periods 

 1 and 3 and to allow the merging of all three periods 

 into one continuous time-series, several parameters 

 were required that could be estimated only with data 

 from Period 2. For this reason, we will start by describ- 

 ing the biomass estimation procedures for Period 2. 



Period 2 The initial biomass estimates obtained for 

 Period 2 (i.e., Leslie estimates of initial biomass in each 

 year, Bq t) may not include the total biomass of armor- 

 head on the SE Hancock Seamount. Instead, the ini- 

 tial estimates may include only the biomass of the 

 fishable stock or that portion of the stock occurring on 

 the summit at night and therefore vulnerable to trawls. 

 The question of whether Bot includes the total popula- 

 tion was addressed by testing the equality of two dif- 

 ferent estimators of annual recruitment to the sea- 

 mount. The first (Rj ) was calculated as the difference 

 between the estimated biomass in 1 year minus the 

 expected biomass surviving from the previous year. 

 Assuming that the catch was taken in a brief interval 

 at the start of the year, this relationship can be ex- 

 pressed as 



Ri,t+i = Bo,t+i - (Bo,t - Ct) e 



(4) 



where B,, t and Bqi+i are the biomass estimates in 

 years t and t-H 1, Ct is the catch in year t, and e "^ is 

 the annual survival rate. The second (R2), is calculated 

 as the proportion of the biomass composed of recently 

 recruited fish: 



R 



2,t+l 



= Bo t + 1 P 



r.t + l I 



(5) 



where Pr,t+i is the proportion of the population com- 

 posed of the cohort recruiting in year t -1- 1 . If Bq, t 

 estimates include the total biomass, then they will be 

 appropriately scaled to Ct, and Ri will equal R2. But 

 if the Bo, t estimates are less than the total biomass, 

 then Ri will be greater than R9. Equality was tested 

 using the statistic 



Z = 



Ri - R9 



VVar(Ri-R2)' 



(6) 



where Z was assumed to be distributed as a normal ran- 

 dom variable. Estimates of Var(Ri - R2) were com- 

 puted as described in the Appendix. 



The fishable proportion of the stock (Pf) was esti- 

 mated in two stages. First, total biomass of the 1980 

 cohort was estimated for each year in 1980-84, when 

 the 1980 cohort represented more than 90% of the 

 total population, by using an age-structured analysis 

 (Megrey 1989) applied to a single cohort. Starting with 

 a known or assumed value of biomass at the beginning 

 of 1985, this analysis sequentially predicts biomass in 

 each preceding year by accounting for catch and 

 natural mortality. If the catch occurs over a short 

 period at the start of the year, total biomass of this 

 cohort in each year (B*t) can be expressed as 



B*to^i = B*toe^' + Ctci 



B*to-2 = B*to-i e^i + Cto-2 



= B*to e^ + Cto-i eM + Cto-2 (7) 



B' 



tO-n 



B 



to 



,nM 



1 Cto- 



i(n-i)M 



i=l 



where B* is an estimate of total biomass at the start 

 of the last year (to) in the time-series (terminal bio- 

 mass), and Cto-i is the catch in year to_i. Second, the 

 proportion fished in each year (P,- , ) was then esti- 

 mated as Bo,t/B* , and mean Pf was then estimated as 

 the average of the five annual estimates. This estimate 

 of Pf, however, was not unique because it depended 

 on B*to, and B*to was chosen arbitrarily because no in- 

 dependent estimate was available. Therefore, the term- 

 inal biomass B*85 (the terminal fishing year was de- 

 fined as 1985 as a later convenience) was estimated 

 along with Pf. Assuming Pf is a constant, the two 

 parameters were estimated by minimizing the weighted 

 sum of squares of the Pf , with weights equal to the 

 inverse of the variance of each Pf t . 



Once the estimate of mean Pf had been obtained, 

 corrected estimates of the initial biomass in each year 

 (i.e., corrected Leslie estimates) were estimated as 



B 



o.t - 



Bo.t 

 Pf' 



(8) 



