704 



Fishery Bulletin 93(4), 1995 



<u 



(0 



cr 



0.015 0.025 0.035 0.045 0.055 



Frequency parameter (c) 

 Figure 3 



» 2 



Map of the function (sin[rf] - sin[c?]) , where c is the 

 true value of 0.0172 day^ 1 and c represents an esti- 

 mate of c. 



ity in zone B was 0.1 because a substantial fraction 

 of the tags in each sample were still recovered in 

 advection area 2 (see Fig. 5). Some of the tags recov- 

 ered in area 1 undoubtedly passed through area 2 at 

 some point as well, further augmenting the amount 

 of information pertinent to estimating the advection 

 in area 2. 



The estimation procedure did not perform nearly 

 as well at high diffusivities as it did at low diffusivities 

 except when the recovery rates were the same in both 

 areas (1.0). In that case the estimates remained unbi- 

 ased and relatively precise — the CE's having dropped 

 rapidly with sample size to less than twenty percent 

 (Fig. 6). The CE's increased dramatically when the 

 recovery rates differed between zones, mostly reflect- 

 ing the corresponding increase in bias (Fig. 7). 



The trends in the CE's also indicate that localized 

 increases in recovery rates may improve the preci- 

 sion of local estimates at the expense of the preci- 

 sion of estimates for the other areas. The CE's in area 

 2 went down with increased recovery rates in zone 

 B, but the CE's in area 1 went up. This effect, how- 

 ever, is largely an artifact of keeping the sample size 

 fixed; increasing the fraction of recoveries near area 

 2 directly decreases the effective sample size for area 

 1 and vice-versa. In reality, local increases in recov- 

 ery rates should add to the number of local recoveries 

 more than they subtract from the number of recover- 

 ies elsewhere, unless the overall recovery rates are very 

 high and there is a great deal of mixing between zones. 



Discussion 



The methods advanced in this article are fundamen- 

 tally different from those cited previously because 



they predict trajectories rather than local abundance. 

 Abundance-based and trajectory-based approaches 

 both assume that tagged and untagged populations 

 move the same way, but they differ in the ancillary 

 assumptions they make. Abundance-based estima- 

 tors, by their very nature, must enumerate a very 

 large number of assumptions regarding processes 

 that could affect the local abundance of the tags. In 

 contrast, the trajectory-based estimators discussed 

 so far disregard everything but velocity. 



It was demonstrated earlier that trajectory-based 

 estimators can produce unbiased estimates of the 

 advection field for the population in general provided 

 that either the tag recovery rates are homogeneous 

 in space and time or the velocity variance is small. If 

 neither of the above conditions are met, the estimates 

 may be biased because faster individuals are more 

 likely to move through regions with different recov- 

 ery rates and may, therefore, be overrepresented or 

 underrepresented in the sample. This is true even if 

 the data are obtained from archival tags. The simu- 

 lation experiments, however, indicated that the bias 

 may not be too severe unless the recovery rates dif- 

 fer a great deal between areas. Moreover, recovery 

 rates generally will not be a factor when the tags are 

 tracked by radio, ultrasonic, or visual means. 



These conclusions suggest that trajectory-based 

 methods are appropriate for many real data sets. In 

 addition, in many cases they may be much more prac- 

 tical than abundance-based methods. Although all 

 of the derivations so far have been in one dimension, 

 it is relatively simple to extend the methodology to 

 include two or three dimensional movements, as is 

 done in the second subsection. Finally, a third sub- 

 section is devoted to discussing the possibilities for 

 adjusting the estimators to account for strongly 

 inhomogeneous recovery rates. 



Practical utility of trajectory-based estimators 



Trajectory-based estimators have several advantages 

 over their abundance-based counterparts. First, 

 whereas trajectory models operate in continuous 

 space and time, tag abundance models are obliged to 

 operate in discrete space and time (one cannot speak 

 meaningfully of the numbers of tags recovered at 

 infinitesimal points). Thus, abundance models suf- 

 fer from truncation errors that occur when different 

 positions are lumped in the same category. If the time 

 and space grid is fine enough, the truncation error 

 will be minimal, but there may then arise the prac- 

 tical estimation problem of having very few observa- 

 tions in any given space-time category. 



A second practical advantage relates to the choice 

 of models. Both trajectory-based and abundance- 



