Ralston et al : An approach to estimating rockfish biomass from larval production 



133 



Pioneer and Ascension Canyons from 1800 h on 21 Febru- 

 ary to 0600 h on 23 February. Where bottom depth permit- 

 ted, discrete depth samples were gathered using 505-pm 

 mesh nets, sampling obliquely in the 0-40, 40-80, 80-120, 

 120-160, 160-200, 200-300, and 300-400 m depth inter- 

 vals. Sampling was arrayed along seven onshore-offshore 

 transects, each composed of three tows conducted at dif- 

 ferent bottom depths, i.e. mid-continental shelf (110 m), 

 the shelf-break (183 m), and well off the shelf (550 m). All 

 samples were preserved in EtOH and after sorting, iden- 

 tifying, and enumerating the larvae in the laboratory, we 

 expressed abundances as the number of shortbelly rock- 

 fish larvae per 1000 m'^ water sampled. 



Spawning seasonality 



At its inception, this assessment was intended to be an 

 application of the fecundity reduction method described by 

 Lo et al. (1992, 1993). However, samples from the Febru- 

 ary and March adult trawl surveys showed that a higher 

 proportion of females had completed spawning in the ear- 

 lier cruise in comparison with the later cruise, an indica- 

 tion that sampling was not representative during one or 

 both of the cruises. Consequently, the fecundity reduction 

 method was abandoned and an alternative approach was 

 devised. Instead, we estimated the seasonal distribution 

 of spawning activity based on the temporal distribution 

 of preflexion shortbelly rockfish larvae in samples col- 

 lected as part of the CalCOFI program from December 

 to April from 1952 to 1984 (see Ahlstrom et al, 1978). We 

 then used this seasonal spawning pattern to expand our 

 estimate of the daily spawning biomass from the short 

 period represented by our 1991 plankton samples to the 

 entire year. 



To estimate the seasonal spawning distribution, we first 

 identified the appropriate samples from CALCOFI station 

 63.55 in the vicinity of Pioneer Canyon by using results 

 from MacGregor ( 1986) as a guide. Plankton samples at this 

 location (Fig. 1) were re-examined and the total number of 

 preflexion shortbelly rockfish larvae were enumerated from 

 Sebastes subsets. These samples amounted to 41 plankton 

 tows (bongo and ring nets) taken in 21 different years. 



Next, we calculated the mean density of preflexion lar- 

 vae for each month, assigned these densities to the mid- 

 point day of the month, and used nonlinear least-squares 

 regression to fit the normal curve to approximate the sea- 

 sonal spawning pattern, i.e. 



N,(t)- 



¥ 



^t-iij' 



where N „, = the estimated density of preflexion larvae 

 on calendar day t (for December t is nega- 

 tive); 

 /J = the expected value of the seasonal distribu- 



tion of preflexion larvae; 

 the standard deviation i 

 and 

 f = a "nuisance" scaling constant. 



a = the standard deviation of the distribution; 

 and 



To determine the seasonal distribution of age-0 larval 

 production that generates the seasonal distribution of 

 preflexion larvae, we assumed that the preflexion larval 

 period has a duration of 15 days (Laidig et al., 1991) and 

 that larvae experience the estimated preflexion mortality 

 rate (see above). As with preflexion larvae, we approxi- 

 mated the seasonal distribution of age-0 larval production 

 with a normal curve, with mean;/f, and standard deviation 

 a^. Given particular values for/^„ and a^^ we calculated the 

 corresponding relative numbers of age-0 larvae produced 

 during each day of the spawning season, and the integrat- 

 ed seasonal distribution of preflexion larvae, i.e. 



N,(t). 



^£iVo«- 



i)e- 



We started the estimation with trial values of //q and a^ 

 and then recursively adjusted the parameter estimates 

 until the mean and standard deviation of the inferred sea- 

 sonal distribution of preflexion larvae (A'',,,) converged on 

 the empirical estimates of u and a . 



Lastly, based on the timing of the larval survey within 

 the long-term mean spawning distribution, total biomass 

 was calculated by simple expansion. Specifically, the 

 midpoint of the 1991 bongo survey was 11 February (i.e. 

 calendar day 42). Consequently we calculated A, which is 

 the proportion of annual spawning under the age-0 larval 

 production curve that occurs from day 41.5 to 42.5. Total 

 population biomass was then estimated by multiplying 

 the estimate of "daily" biomass by Ilk. This calculation im- 

 plicitly assumes that the seasonal progression of spawn- 

 ing has been stable over years. Therefore, the sensitivity of 

 the biomass estimate to a violation of this assumption was 

 evaluated by profiling over a range of values for the mean 

 date of spawning (;/,,), which has a substantial effect on A. 



Precision of the biomass estimate 



The determination of total biomass requires the estimation 

 of numerous statistical relationships, each with its own 

 parameter set. The results of fitting these functions were 

 then combined algebraically to produce the final biomass 

 estimate. We calculated the precision of the final biomass 

 estimate by using the delta method (Sober, 1982. p. 8), i.e. 



v\g(e)\ = Y^v\e,\\ — \ +2^^cov[0„0,]^ 



he, de, 



where g(0) = the algebraic combination of functions used 

 to produce the final biomass estimates; and 

 6 = the full set of estimated parameters. 



Application of this method requires estimates of variances 

 for each parameter, covariances among parameters, and 

 partial derivatives of estimated biomass with respect to 

 each parameter (dB/dO). Partial derivatives of the final 

 biomass estimate with respect to the parameters were cal- 

 culated numerically by using central differencing by per- 



