Porch. Estimating velocity and diffusion from tagging data 



707 



tag, then the estimated advection field should be rela- 

 tively accurate. Otherwise the estimates may be bi- 

 ased, either because the velocity variance is large or 

 because the advection model is poorly specified. 



The simulation results revealed that the estima- 

 tors were essentially unbiased when the low level of 

 diffusivity was used in generating the data. The root- 

 mean-square displacements associated with this level 

 of diffusivity were on the order of two percent of the 

 advective displacements. This suggests that random 

 displacements of at least two percent, and perhaps 

 much larger, can be considered "negligible." Simula- 

 tions that are tailored to specific situations are rec- 

 ommended to determine more accurately the toler- 

 ance of any particular application. 



Extensions to multiple dimensions with 

 boundaries 



The mathematical derivations in this paper were 

 developed in one dimension and without regard to 

 barriers, primarily to simplify the presentation. It 

 may sometimes be convenient to describe tag mo- 

 tion in this manner, such as when the tags are em- 

 bedded in a major ocean current. Otherwise, the 

 methods can easily be extended to accommodate more 

 complicated scenarios. 



Multiple dimensions can be incorporated by con- 

 structing dimension specific components in the ob- 

 jective function, e.g. 



PxTt 



(y,-yf 



PyT, 



where P and fi are the diffusivity parameters in the 

 two-dimensional space spanned by the coordinates x 

 and v. The predicted positions are then obtained by 

 integrating the differential equations describing the 

 advection field in the two-dimensional space. Al- 

 though multidimensional equations of motion are 

 typically more difficult to solve analytically than their 

 one-dimensional counterparts, they can always be 

 integrated numerically. The other alternative is to 

 use discrete approximations, in which case the mo- 

 tion along each dimension can be treated indepen- 

 dently and no further modifications to the methods 

 are necessary. The x and y positions of each tag are 

 then predicted by separate parameterizations of 

 Equations 11 and 12. 



Barriers to tag motion, such as coastlines or ther- 

 mal fronts, tend to preclude analytical solutions to 

 the equations of motion but are relatively easy to 

 handle in a numerical context. The position of each 

 tag is updated at regular intervals by numerically 

 integrating the velocity equations. When the tag 



encounters the barrier, it reacts according to some 

 prescribed behavior pattern. Subsequently, the nu- 

 merical integration proceeds as described earlier. The 

 appropriate behavior prescription depends on the type 

 of barrier and the nature of the tagged object. Some 

 common choices include reflecting, absorbing, and stick- 

 ing barriers — but the suite of possibilities is endless. 



Extensions to incorporate variable recovery 

 rates 



This part of the article addresses the possibility of 

 modifying trajectory-based estimators to accommo- 

 date situations where the diffusive displacements are 

 not negligible and the recovery rates are not homo- 

 geneous in time and space. The matter essentially 

 condenses to determining the theoretical probabil- 

 ity density of the position of recovered tags so that 

 an appropriate maximum-likelihood solution can be 

 formulated. The predictor is not an issue because it 

 is, by nature, independent of the recovery rates. 



The simplest way to approach the problem is in 

 terms of sampling strategies. Each recovered tag can 

 be thought of as a nonrandom selection from the 

 underlying probability density of the tagged popula- 

 tion. Tags recovered at any specific location x are 

 therefore misrepresented in the sample by a factor 

 P R (x), which is the probability of recovering a tag at 

 x. For example, if the underlying probability density 

 for the in situ positions of the tags was the normal 

 distribution with variance fiT, then the adjusted ob- 

 jective function would be 





(20) 



The correction factor P R can be any measure of the 

 relative likelihood of a tag arriving at any given po- 

 sition. In principle, it can be expressed as a function 

 of the parameters of the underlying models of the 

 population dynamics and estimated as part of the 

 overall parameter search. Towards this end, it is 

 important to recognize that the only relevant recov- 

 ery processes are those that vary among tags. For 

 example, if faster individuals were not significantly 

 more likely to shed their tags than were slower indi- 

 viduals, then tag-shedding would be irrelevant to 

 trajectory-based models regardless of its magnitude. 

 The same would not be true for abundance-based 

 models because tag-shedding would help to deter- 

 mine the total number of recoveries. Thus it should 

 be possible to adjust trajectory-based estimators so 

 that there are fewer parameters than those required 

 by abundance-based estimators. 



