696 



Fishery Bulletin 93(4). 1995 



of the measure's predictor. To illustrate, consider a least- 

 squares objective function whose measure is the num- 

 ber N of tags recovered in each area a and season s: 



t,o + T, 



^^(N as -N as f 



(5) 



The predictor, N, is a function of the parameters of 

 the underlying population dynamics and advection 

 models. 



In theory, the parameter estimates that minimize 

 Equation 5 are unbiased if the measure (N) is nor- 

 mally distributed with constant variance and the 

 models behind the predictor are correct. It is gener- 

 ally recognized, however, that the probability den- 

 sity of recoveries is nonnormal and the variance in 

 N may differ widely among areas and seasons, both 

 of which imply that the least-squares solution is in- 

 appropriate. For this reason, most of the recent lit- 

 erature has favored maximum-likelihood solutions 

 based on the multinomial distribution instead. 



The condition of the predictor ( N ) is more diffi- 

 cult to assess because it embodies a suite of assump- 

 tions regarding all relevant aspects of the popula- 

 tion dynamics. The local abundance of tags may be 

 affected by processes common to both the tagged and 

 untagged populations (e.g. natural and fishing mor- 

 tality) as well as by processes that are unique to the 

 tagged population (e.g. tag-induced mortality, tag 

 shedding, and failures to report recovered tags). 



It would be advantageous to develop a predictor 

 that does not need to account for all the complex pro- 

 cesses affecting tag recoveries, but it is clear that a 

 measure other than abundance must be used. The 

 trajectories of individual tags, which can be predicted 

 from their velocity history alone, is one such mea- 

 sure. Tag recovery rates are relevant to trajectories 

 only in the sense that they determine those most 

 likely to be represented in the sample. That is, re- 

 covery rates dictate the probability density of ob- 

 served (recovered) trajectories, but not the predic- 

 tor. This point, though subtle, has important implica- 

 tions with respect to relaxing the assumptions required 

 to produce unbiased estimates of the advection field. 



Consider that the expected position of a tag after 

 liberty time T follows from Equation 2: 



l, n +T, 



E[x lT ] = x l0 + \u(x,,t)dt. 



(6) 



The expected position of a recovered tag under the 

 same conditions is 



E R [x lT ] = x l0 + \u(x t ,t)dt + 



Er 



u'(x t ,t)dt 



where the subscript R indicates that the expectation 

 includes recovered tags only. The second integral 

 dropped out of the unconditional expectation in Equa- 

 tion 6 because u' is, by definition, a random variable 

 with mean equal to zero. The same would not gener- 

 ally be true of the expectation of recovered tags be- 

 cause some vectors of u ' may be more likely to be 

 recovered than others — changing the probability den- 

 sity in some unknown way. 



Suppose there exists an objective function 0[x,x] 

 (maximum likelihood or otherwise) that can produce 

 unbiased estimates of E[x lT ] from a random sample 

 of all potential trajectories x jT . (The construction of 

 this function will be discussed later.) The same ob- 

 jective function will also produce unbiased estimates 

 from a random sample of recovered tags provided 



E f 



tio+Ti 

 \u'{x t ,t)dt 



(7) 



This constraint is satisfied if either u ' is everywhere 

 identically zero or the probability of recovering a tag 

 is independent of its velocity. The latter condition is 

 effectively equivalent to assuming that the processes 

 that influence tag recovery are homogeneous in space 

 and time. It satisfies Equation 7 because it implies 

 that the relative likelihood of observing any given 

 displacement depends solely on the probability den- 

 sity of u' . By definition, the expectation of u' at ev- 

 ery point is zero and therefore the expectation of the 

 integral sum of u' is also zero. 



It has been shown that, subject to Equation 7, tra- 

 jectory-based estimators can provide unbiased esti- 

 mates of the population advection field without re- 

 covery rates having to be considered. One can imag- 

 ine many practical situations where Equation 7 

 would be approximately satisfied. The spatial and 

 temporal distribution of recovery rates would not 

 normally be a significant factor in experiments in- 

 volving radio-tracked drifter buoys or ultrasonic tags. 

 Similarly, variations in the velocity of individuals 

 might be expected to be small compared with the 

 average velocity of a population undergoing a sea- 

 sonal spawning migration. Where Equation 7 is not 

 met, however, tags moving at different velocities may 

 not be equally represented in the recovered sample. 



