666 



Fishery Bulletin 97(3), 1999 



Table 3 



Treatment of missing observations in the NMFS spring and 

 fall surveys to estimate abundance on the U.S. side and 

 the Canadian side of unit areas 5Zj and 5Zm. c = Cana- 

 dian side of stratum; u = U.S. side of stratum; M and M = 

 used in multiplicative model; M = missing observations 

 estimated from multiplicative model; 5Z = 5Z stratum mean 

 used to fill in missing observations; = missing observa- 

 tions assigned a value of 0. 



Stratum 



Spring 



Fall 



1991.75, for ages 1-8, a, was calculated by using vir- 

 tual population analysis with a three-month time 

 period, i.e. t-0.25, according to the following algo- 

 rithm. The annual instantaneous natural mortality 

 rate, M, was assumed constant and equal to 0.2. In 

 all equations N refers to numbers offish and, if pre- 

 ceded by a subscript letter, indicates the source of 

 the loss so that ^N is the number dying from fishing, 

 ^A'^ is the number dying from natural mortality, and 

 i^ is the number lost or gained from migration. If 

 not preceded by a subscript, A^ refers to population 

 abundance. The method assumes an exponential 

 decay model: 



^.,a=A^.v«,.. 



where the fishing mortality, F^,^. for ages 1 to 7 is 

 obtained by solving the catch equation using a New- 

 ton-Raphson algorithm with the three-month com- 

 mercial catch-at-age data, pN ,^- 



Ny.a = 



PN,^JF^_„+M)t 



Fyjn- 



~IF,„+M>t 



adjacent stratum section of the same stratum. When 

 both strata sections of a stratum had missing obser- 

 vations, the age composition from an adjacent stra- 

 tum was used. Haddock abundance at age on the 

 Canadian and U.S. sides of the ICJ line were obtained 

 by applying the age compositions of strata and strata 

 sections to their respective total abundance and then 

 summing the results within each jurisdiction. A ra- 

 tio of relative abundance for the Canadian side was 

 calculated by dividing the estimate on the Canadian 

 side by the sum of the estimates on Canadian and 

 U.S. sides. 



Instantaneous rates of net migration 



The availability of fisheries statistics, since 1985, at 

 a resolution sufficient to be summarized with respect 

 to the ICJ line, in conjunction with the relative abun- 

 dance information described above, made it possible 

 to derive estimates of net migration rates. Catch-at- 

 age by quarter which was required for this analysis 

 is reported in Gavaris and Van Eeckhaute ( 1997). 



Net migration rates were estimated by using a 

 model that follows from one originally proposed by 

 Beverton and Holt ( 1957 ). Terminal population abun- 

 dance for each cohort at the beginning of 1994 for 

 the 1986 to 1993 year classes and at age 8 for the 

 1968 to 1985 year classes was taken from Gavaris 

 and Van Eeckhaute ( 1994 ). Population abundance in 

 all of 5Zj,m at time y, expressed in units of years, e.g. 



The total number of fish dying due to natural mor- 

 tality in each quarterly period, y, was obtained by 

 applying the quarterly rate of natural mortality to 

 the average population abundance during the period: 



M^yM 



MtN, 



where N, the average population abundance during 

 the period, is defined as 



iV„ 



.v..(] 



N„Jl 



-tF,„+M>t 



{F,.„+M)t 



DFO surveys were conducted between 10 Febru- 

 ary and 19 March, whereas NMFS spring surveys 

 were conducted from 23 March to 26 April. The NMFS 

 fall surveys were conducted between 3 October and 

 25 October. The distributions of haddock obtained 

 from the spring and fall survey results were consid- 

 ered to be representative of 1 April and 1 October, 

 respectively. For each year, six-month net migration 

 rates were calculated for the spring-summer period 

 (1 April-30 September) and the fall-winter period 

 (1 October-31 March). 



Since proportions of haddock on the Canadian side 

 of the ICJ line, as mdicated from DFO and NMFS 

 survey results, were available for only two points in 

 time, spring and fall, subsequent calculations were 

 done on a half-year basis to coincide with the survey 



