Abstract. Several new models 



are developed to estimate the velocity 

 and diffusion of a population from tag- 

 ging data. The new estimators apply 

 the inverse principle to the individual 

 trajectories of recovered tags rather 

 than to their local abundance. These 

 models require fewer assumptions and 

 less information than do published 

 abundance-based methods. Techniques 

 are presented for a variety of circum- 

 stances, and both discrete and continu- 

 ous parameterizations of the velocity 

 field are included. The sensitivity of the 

 estimators to violations of the assump- 

 tions was examined numerically by us- 

 ing stochastic simulations. The results 

 suggest that the estimators are fairly 

 robust but may fail under certain con- 

 ditions. Extensions to accommodate 

 these situations are discussed. 



Trajectory-based approaches to 

 estimating velocity and diffusion 

 from tagging data 



Clay E. Porch 



Southeast Fisheries Science Center 



National Marine Fisheries Service, NOAA 



75 Virginia Beach Drive, Miami, Florida 33149 



Manuscript accepted 17 April 1995. 

 Fishery Bulletin 93:694-709 ( 1995). 



Tagging experiments have often 

 been used to delineate animal move- 

 ments. Historically, many of the 

 analyses were limited to graphical 

 portrayals of apparent migration 

 routes and simple measures of the 

 net swimming speed. Beverton and 

 Holt (1957) introduced more rigor- 

 ous treatments of tagging data 

 based on the pioneering work of 

 Skellam (1951). They postulated 

 that the Fickian diffusion equation 

 would be an adequate model for the 

 dispersion of fish due to random 

 motions and developed a technique 

 for estimating the diffusion coeffi- 

 cient from tagging data. Subse- 

 quently, Jones (1959, 1976) devel- 

 oped a simple estimation procedure 

 that distinguished the diffusion and 

 net drift of returned tags. Saila and 

 Flowers (1969) proposed using a 

 special case of the advection-diffu- 

 sion equation (Fickian diffusion 

 with constant velocity) to model fish 

 migration and developed a numeri- 

 cal technique to estimate the diffu- 

 sion coefficient and net drift. More 

 recently, Sibert and Founder ( 1994) 

 advocated the use of a more general 

 form of the advection-diffusion 

 equation that allows for mortality 

 and discrete changes in velocity 

 among areas. They also developed 

 a new estimation procedure based 

 on fitting numerical predictions to 

 the observed distribution of recov- 

 ered tags. Similar methods have 

 been applied to the movements of 

 passive tracers in the ocean by 



Fiadero and Veronis (1984) and 

 Wunsch(1989). 



Beverton and Holt (1957) recog- 

 nized that the solutions to advec- 

 tion-diffusion equations are greatly 

 complicated by heterogeneous dif- 

 fusion rates and irregular boundary 

 conditions (e.g. coastlines). They 

 suggested replacing the diffusion 

 equation with a system of area-spe- 

 cific equations linked together by 

 transfer coefficients that measure 

 the movement across the boundary 

 of adjacent areas (box models). This 

 simple abstraction, as well as oth- 

 ers like it, has received considerable 

 attention in recent years, and a 

 number of papers have dealt with 

 estimating the transfer coefficients 

 from tagging data (Beverton and 

 Holt, 1957; Sibert, 1984; Hilborn, 

 1990; Deriso et al., 1991; Hampton, 

 1991; Schweigert and Schwarz, 

 1993; Kleiber and Fonteneau, 1994; 

 Salvado, 1994). In principle box 

 models are not very realistic be- 

 cause they assume that movement 

 within and among boxes occurs in- 

 stantaneously, but in practice they 

 may approximate the dynamics well 

 enough to be useful. 



All of the aforementioned proce- 

 dures (except Jones's) estimate the 

 movement of a population from the 

 local abundance of recovered tags. 

 In contrast, the methods developed 

 in this paper estimate movement 

 from the trajectories of recovered 

 tags. Strictly speaking, the new 

 models address the advection (ex- 



694 



