salmon catcli hy time period within a season for a 

 parlirular lisliery, tlie number of tish examined 

 for marks, and the number of marked fish re- 

 coxered. In addition to obtaining an estimate of the 

 cat<'li of a given age group of hatchery fish, we 

 are estimating the percentage contribution of 

 hatcliery fish to tlie total catch of that age group. 

 This makes it necessary to subdivide the catch by 

 age group on the basis of an estimated age com- 

 position. 



For a ])articular fishery (e.g., Washington 

 ocean sport fishery at Westport) let: 

 Ci= catch during i"' period 



ni = iuimber of fisii exannned for marks during 

 i"" period 

 nijk = nundjer of marked fish with the k"' kind of 

 mark and of the j"' age group recovered 

 diu-ing the i"" period 

 Pijk= I'l'oportion of Ui having k"' kind of mark 

 and of the j'" age group 

 = n,)k/'ni 

 Tlien an estimate of the catch of fish with the 

 k'" kind of mark and of the j'" age group for the 

 i"' period is: 



mi)k=pi,kC| (1) 



An estiniate of the catch of all marketl fish of 

 the j'" age group for the i'" jjeriod is: 

 mii. = c,2pijk 



k 



= 2m,jk 



k 



and an estimate of the catch of all marked fisii 



of all age groups for the i'" period is: 



m, , = 2ni,j. 

 i 

 Tlie estimated catch of luimarked fish during 



the i'" period is then: 



u, = Ci— m,.. 

 If gij is the j)roportion of the samjjle of un- 

 marked fish belonging to the j"" age group for the 

 i'" period, then an estimate of the catch of un- 

 nuirked fish of the j"' age group for the i'" period is: 



Uij = gijU| 



Hence, the estimate of the total catch (marked 

 and unmarked) of the j'" age group for the i'" 

 period is: 



Cij = ^iu + m,j. 

 and tlie estimated seasoiud catch of marked 

 and unmarked fish of the j'" age group is: 



c, = 2c„ " (2) 



i 



Now, if we designate a particular hatchery 

 evaluation mark (e.g., adipose-right maxillary) 



by setting k=l in ecpiation (1), then an estimate 

 of the seasonal catch of Columbia River hatchery 

 fish of the j'" age group having that mark is: 



m.j, = 2pij,c, (3) 



i 



If, for example, we are interested in marked 

 hatchery fish (adipose-right maxillary mark) of 

 the 1961 brood, then their estimated catch in 

 1963 as 2-year-old fish is: 



3m 21 = 2(3Pi2i) (3C,) 

 i 



where the presubscript denotes the last digit of 

 the sampling year. Similarly, the estimated 

 catch in 1964 as 3-year-old fish is: 



4m .31= 2 (4P 131) (4C1) 



i 

 ami the estimated catch in 1965 as 4-year-old 

 fish is: 



5m.41=2(5Pul) (5C1) 

 i 



and similarly for still later years of captau'e. 



An approximate variance estimate for m.ji is: 



V(m.),)=C^S(w^p,j,/n,) 



' +V(C)^(2w,pu,)'-2w^p,H/n^ (4) 



where C=2ci, w, = Ci/C- and V(C) is the variance 



i 

 of the estimated seasonal catch. If the catch, C, 

 is known, then only the first item in (4) contributes 

 to the variance estimate. 



Now if ai'ji is the expected proportion of 

 Columbia River hatchery fish of the j*" age group 

 in samplii\g year "a," which is marked with a 

 particular mark (e.g., let k= 1 for adipose-right 

 maxillary), then an estimate of the seasonal 

 catch of hatchery fish (marked and immarked) of 

 the j"" age group in sampling year "a" is (from 

 equation 3) : 



aHj = am.j,/ar)i (5) 



Hence, if, as in the example above, we are in- 

 terested in the contribution of the 1961-brood 

 hatchery fish, then an estimate of their catch 

 (marked and unmarked) in 1963 as 2-year-old 

 fish is: 



3H2 = 3m.2l/3r21 



Likewise then, the estimated catch in 1964 as 

 3-year-old fish is: 



4H3^4m 3]/4r3i 



and sinuhuly for the 1965 and 1966 sampling 

 years. 



An alternative procedvn-e for estimating the 

 seasonal catch of hatchery fish of the j'" age 

 group in sampling year "a" is: 



CONTRIKfTIUN OF CULUMBIA RIVER HATCHERIES TO FALL CHINOOK SALMON HARVEST 



369 



