FISHERY BULLETIN: VOL. 71, NO. 2, 1973 



here), both C and the inshore catch may be 

 identified by origin (e.g., Worlund, Wahle, and 

 Zimnier, 1969; Johnson, 1970). The i)OssibiHties 

 of 1) C known but inshore catch unknown with 

 respect to origin or 2) offshore fishing present 

 but inshore fishing absent are too remote for 

 consideration here. Our two general cases — 

 C known and unknown — are therefore defined 

 and the inshore catch is implicit in £• . Figure 

 1 contrasts the extinction of a smolt class 

 with and without offshore fishing. 



It is evident from Equations 1-5 that the 

 .s^- and m are confounded, and no unique solution 

 exists. This fact and the existence of offshore 

 salmon fisheries led to development of the 

 indirect approach. 



ESTIMATING MODELS 



Three existing models are reviewed in terms 

 of biomass computations, bias in estimated 

 mortality schedules is considered, and a new 

 estimating scheme is introduced. 



Ricker's (1962) Model 



This model is based on convincing if indirect 

 evidence (not direct measurements) that most 

 natural mortality during the ocean life of 

 sockeye salmon occurs well before the fish are 

 large enough to be recruited to an offshore 

 fishery. In the context of Equations 6-8 with 

 F = C = 0: 



= M^>M^. 



(9) 



In Equation 9, M^ is the average monthly rate 

 of natural mortality during t^. — f„, /„ being 

 the date when (say) half the smolt class enters 

 the sea in 1 year and ty the date of potential 

 recruitment offshore the next year (say). Parker 

 (1968) demonstrated from direct marking/re- 

 covery of pink salmon that natural mortality is 

 highest during the first few weeks of ocean 

 life. To my knowledge, salmon biologists all 

 agree that Mi > M-y. 



In the absence of direct measurements, 



Ricker's model assumes ni = and .s-2 = 1.0 

 on the Ex jacks (Equations 1 and 5); it treats 

 El + Eo = ^E^ as a single escapement at 

 time / = T, with the following result from 

 Equation 5: 



Zi = Mi 



Mri = ^2t/ 



-\ni^E^INo)l{T-t^). 



(10) 



In Equation 10 the caret symbol (*) denotes, 

 an estimate, and the subscripts L and U 

 denote lower and upper limits, respectively. 



Because M^y < G (the estimated growth 

 coefficient) after I = t^ for most age and 

 maturity groui)s of commercial size in offshore 

 waters, Ricker (1962) concluded that offshore 

 fishing is biologically wasteful. Those biomass 

 computations assume 100% availability to an 

 offshore fishery, however, and overestimate 

 minimum losses. Ricker (1964) later mentions 

 the availability in connection with the growth- 

 mortality balance in pink and chum salmon and 

 computes weight losses from offshore fishing 

 as a fraction of maximum possible yield (inshore 

 fishing only) for any fixed spawning escapement 

 required for reproduction. Biomass calculations 

 of Parker (1963) and Fredin (1964) also assume 

 full availability. Although the evidence indicates 

 offshore fishing reduces total yield (see also 

 Cleaver, 1969; Henry, 1971, 1972), I emphasize 

 that schedules of growth, mortality, and avail- 

 ability must be combined in order to assess 

 the impact of existing or potential offshore 

 fisheries: management restrictions, fleet size, 

 and bad weather always prevent continuous, 

 complete availability of a stock offshore. 



Ricker's model (Equation 10) was applied 

 only to situations where F = C = 0. The 

 direction if not the magnitude of bias in the 

 mortalitv schedule is known from evidence 

 already cited, i.e., 3/, < Mi and M2 > M-- When 

 an offshore fishery exists (F > in Equation 

 4) and Zi = Mi as noted for coho salmon, 

 Equation 10 may be used as the estimator, 

 Z* = Zi = Zo, and the bias defined in terms 

 of known data and the unknown maturity 

 parameter, ni. Absolute bias (B) is defined as: 



B = estimate minus parameter value. (11) 



516 



