aH,= (,M.]./aqi.) + 2:aM.,, 



(6) 



wliere aClji is the expected marked (adipose- 

 right maxillary) to unmarked ratio. The first 

 term in tliis equation is an estimate of the catch 

 of unmarked hatchery fish, and the second term 

 is the estimated total catch of marked hatchery 

 fish (summed over the five marks identifying 

 hatchery fish). 



The percent contribution of luitchery fisli to 

 tlie catch of the j"" age group in year "a" is 

 tiien (from equation 2): aPj=100(aHj/aC.j). 



Equations (5) and (6) are the basic eciuations 

 for estimating the catch of Columbia River 

 hatcliery-reared fall chinook salmon of a given 

 age in a given year. The numerator, aM.ii. hi 

 etiuation (5) is the estimated number of marked 

 fisli (adipose-right nuixillary) in the catch at a 

 l)articular fishery-port combination. As stated 

 before, the denominator, ^Vj], is the ex|)ected 

 |)r()|)ortion that a^I.ji is of the total catch of 

 lialchery fisli of the same age grouj). Granting 

 certain assumptions (set forth in the following 

 section), tiie proi)ortion of adipose-right maxillary 

 marked fish in the hatchery releases is an estimate 

 of ai'ji- Hence, if r is the jirojiortion of marked 

 fish in the 1961 brood hatcherv releases, we set 



3r2i — 4r3i = 



.ii'41 — fii'.ii 



So, we assume that the expected proportions 

 of marked age 2 hatchery fish in 1963, age 3 

 fish in 1964, etc. are identical and equal to the 

 pro|)ortion of marked fish in tlie releases of tlie 

 1961 brood hatchery fisli. 



The estimated proportion of marked fish in the 

 total release of each brood year from the 12 

 hatcheries is based on estimates of the proportion 

 of marked fish in the release and the size of release 

 at each of the hatcheries. These latter estimates 

 were obtained from a sampling ])rocedure at the 

 time of release at each of the hatcheries. If N„ 

 is the total number of fisli released from a hatchery, 

 then the number of fish removed by the sampling 

 device at the first stage of sampling is : 



N 



l = Sl 



N„ 



^\'Ilere Si is the actual ])roi)ortion of the total 

 number of fish removed by the sampling device. 

 Tliese fish, Nj, were then sampled again. If So is 



the actual proportion of the total number of fish, 

 Ni, removed by the sampling device during tlie 

 second stage of sampling, then the number of fisli 

 which comprise the final sami)le is: 



Hence, if s is an estimate of the proportion of fish 

 removed by the sampler during a single stage 

 (i.e., s is an estimate of both s, and So), then an 

 estimate of the numbers of fish released, No, is: 



N = N,/s2 



The estimated proportion of fish removed by the 

 sampling device was obtained from a number of 

 trials using known numbers of fish, and the number 

 of fish removed by the sampling device, N2, was 

 counted and sorted into the marked and un- 

 marked groups. Thus, if M is the count of marked 

 fish, then the estimated proportion of marked 

 fish in the release is simply: 



r=M/N2 

 If N, and r, are the above estimated quantities 

 for the i'" hatchery, then the estimated proportion, 

 r, of marked fish in the total release from all 

 hatcheries is: 



r=2:w,r, 



1 



where w,= N,/2N, 



An estimate of the nuirked to unmarked ratio 

 for the hatchery releases is similarly obtained. 



ASSUMPTIONS 



Several assumptions are inherent in the fore- 

 going methotl of estimating the contribution of 

 hatchery-reared fall chinook salmon to the iisher- 

 ies. Three basic assunqitions are: 



1. A marked fish is identifiable as a marked 



fish thr(Uighout its life. 



2. All observed chinook salmon having the 



kind of mark used on the hatchery- 

 reared fish are indeed hatchery fish. 



3. Chinook salmon are correcth' aged from 



scale examinations and information on 

 size of fish anil date of capture. 



370 



f.S. FISH AM) WILDLIFE SERVICE 



