Eveson et al.: Incorporating fishery observer data into an integrated catcti-at-age and multiyear tagging model 495 



in the observed component of the fishery are sampled 

 for length or age, but no fish from the unobserved com- 

 ponent are sampled. The catch monitored by observers 

 is assumed to be representative of the total catch (i.e., 

 catches from the observed and unobserved components 

 have the same expected age distribution). If the expected 

 catch-at-age distribution differed between the two com- 

 ponents, then separate age information would need to 

 be available for each component and the catch likelihood 

 presented below would need to be modified. 



The basic assumptions common to all multiyear tag- 

 ging models, as summarized in Pollock et al. (1991), are 

 also required for the BPO model. The most important of 

 these are: 1) tagged and untagged fish are thoroughly 

 mixed throughout the population of interest, 2) the fate 

 of each fish is independent of the fate of other fish, 3) 

 all fish of a given age class have the same survival and 

 capture probabilities, and 4) there is no tag shedding 

 or tag-induced mortality. If tag shedding or tag-induced 

 mortality, or both, exist at non-trivial levels (i.e., as- 

 sumption 4 is not met), then additional parameters 

 and potentially additional data need to be introduced 

 to account for them. Failing to do so will lead to biased 

 parameter estimates and overly optimistic estimates of 

 their precision. If any of assumptions 1 to 3 is violated, 

 then the variance of the tag return counts will be un- 

 derestimated by the model. Similarly, if assumption 2 

 or 3 is violated, the variance of the catch numbers will 

 be underestimated. Extra variability, or overdispersion, 

 in the tag return and catch data is discussed in the 

 next section. 



Assumption 1 implies that newly tagged fish are mixed 

 throughout the population immediately after tagging. 

 This mixing can be difficult to achieve in practice, espe- 

 cially when the population has a widespread geographi- 

 cal distribution or tagging occurs in a limited area of its 

 distribution. Hoenig et al. (1998b) showed how delayed 

 mixing of newly tagged fish can be incorporated into a 

 Brownie model by allowing these fish to have a different 

 fishing mortality rate in the year of tagging than that 

 of previously tagged fish. In our application of the BP 

 model to southern bluefin tuna (SBT, Thiinnus maccoyii) 

 data in Polacheck et al. (2006), we allowed for initial 

 nonmixing with this approach. Only the tag-recapture 

 component of the model needed to be modified. It would 

 be straightforward to modify the tag-recapture compo- 

 nent of the BPO model in an analogous manner in situ- 

 ations where modification was considered necessary. 



Before proceeding, we introduce the notation that 

 will be used throughout this study. The data required 

 by the model are 



N^ = the number of tag releases of age a fish from a 

 particular cohort; 



R"^ ^ = the number of tag returns from fish that were 

 tagged at age a and recaptured at age / in the 

 observed (o) component of the fishery; 



R"^ ^ = the number of tag returns from fish that were 

 tagged at age a and recaptured at age i in the 

 unobserved (u) component of the fishery; and 



C° = the estimated number of age i fish from the 

 cohort of interest caught in the observed (o) com- 

 ponent of the fishery. 



The model parameters assumed to be known are 



6, = the proportion of fish from the cohort of interest 

 caught in the observed component of the fishery in 

 year i; 

 J)-, = the variance of the aging error for C^. 



The model parameters to be estimated from the data 



M, = the instantaneous natural mortality rate for age i 

 fish; 



F, = the instantaneous fishing mortality rate for age ; 

 fish; 



Pj = the population size of the tagged cohort at the age 

 of first tagging (assumed to be age 1 for conve- 

 nience); and 

 A, = the tag reporting rate for fish captured at age i in 

 the unobserved component of the fishery. 



In addition, the annual survival rate (S, ) and exploita- 

 tion rate (w,), respectively, of an age / fish, are defined 

 to be 



S,=exp(-(F,+M,)); 



(1-S, 



F.+M, 



Note that because only a single cohort offish is being 

 considered, age and year can be used interchangeably 

 in the above definitions. If more cohorts were added to 

 the model, it would then be important to distinguish 

 whether the parameters vary by year, by age, or both. 

 For example, A may vary by year, M by age, and F by 

 both. Because the age distribution of the catch is as- 

 sumed to be the same for the observed and unobserved 

 components, d would vary with year, not age, when 

 there is more than one cohort (i.e., the probability of 

 a fish being caught in the observed component of the 

 fishery in year i would be the same for all ages within 

 the year). If the age distribution of the catch was al- 

 lowed to differ between the observed and unobserved 

 components, then d would need to vary with both year 

 and age, but it would not be estimable unless informa- 

 tion was available about the age distribution of the 

 unobserved catches. 



First consider the tag-recapture component of the 

 model. The probability of a fish, tagged at age a, being 

 caught in the observed component of the fishery at age 

 i, and having its tag returned, is 



Pa, 



SjUj i = a 



S,Sa--S,-iii, i>a 



(1) 



