Somerton and Kobayashi: Inverse method for mortality and growth estimation 



371 



cumulatively summing across all lengths. Individual 

 lengths within a sample were chosen by determining 

 which category in the cumulative length probability 

 distribution just exceeded the value of a generated 

 uniform random number. 



Biased length-frequency samples were generated by 

 simulating the sampling of the model population by 

 using a plankton net, which allowed zero extrusion and 

 produced capture probabilities (Pc.i) described by an 

 inverse logistic function: 



Pc.l = 1 



1 -H 9.00 X 10^ e -2-611 



(3) 



where 1 is length (in mm) [Fig. 2a; parameters in Eq. 

 (1-3) were chosen arbitrarily and were not intended 

 to represent any particular species or sampling gear]. 

 Samples were drawn by using the same procedure as 

 used for unbiased samples except the population length 

 distribution was multiplied by the capture probabOities 

 (Fig. 2b). 



Ageing samples were generated similar to length- 

 frequency samples, but after each length was drawn, 

 an associated age was also drawn by using the cum- 

 ulative age probability distribution at each length and 

 an additional uniform random number. The sample 

 sizes used in the simulations [1000 length-frequency 

 samples, 300 ageing samples, 600 selection samples, 

 with the biased sample size set equal to the unbiased 

 sample size x the average probability of capture deter- 

 mined from Equation (3)] were arbitrarily chosen but 

 were similar to those used in Somerton and Kobayashi 

 (1989, unpubl. data). 



Unbiased length-frequency samples for these three 

 types of data were generated by simulating the ran- 

 dom sampling of a larval fish population (Fig. 2a) with 

 a constant daily recruitment, a constant instantaneous 

 daily mortality rate (M) of 0.20, and a length distribu- 

 tion at each age conforming to a normal probability 

 distribution. Mean (It) and variance (Var(lt)) of 

 length-at-age were chosen, for simplicity, to be linear 

 functions of age: 



It = 10.00 H- 1.50t, and 

 Var(lt) = 2.50 + 0.2&t, 



(1) 



(2) 



where t is age (in days) and It is length (in milli- 

 meters). Samples were drawn from the cumulative 

 length probability distribution of this population that 

 was constructed by dividing each of the population 

 length-frequencies by the total sample size, then 



Growth simulations 



Growth simulations examining the performance of 

 IMMAGE consisted of 1000 repetitions of the follow- 

 ing sequence. First, a biased ageing sample and a selec- 

 tion sample were generated. Second, capture prob- 

 abilities were estimated from the selection sample by 

 fitting an inverse logistic function of length, using 

 nonlinear regression, to the ratios of the biased to the 

 unbiased length-frequencies. Third, initial parameter 

 estimates for the growth process model [Eq. (1) and 

 (2)] were obtained from the ageing sample by fitting 

 straight lines to length-at-age and variance of length- 

 at-age. Fourth, the unbiased length distribution of each 

 age-class was estimated as a normal distribution with 

 mean and variance predicted from Eq. (1) and (2) 

 evaluated at the current parameter estimates. Fifth, 

 the biased length-frequency distribution of each age- 

 class was estimated by multiplying the unbiased length- 

 frequency distribution by the estimated capture prob- 

 abilities. Sixth, parameter estimates for Eq. (1) and (2) 



