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Fishery Bulletin 93(4), 1995 



Mean-square dispersion is defined as the average of 

 the squared deviations of the tags from their expected 

 positions. It is usually expressed as a function of 

 elapsed time T: 



v 2 

 XiT 



1 " 



i, T ) 



(17) 



i=i 



where x lT is the predicted position given the esti- 

 mated advection field, and n is the number of tags in 

 the sample. The absolute diffusivity is defined as one 

 half the time rate of change of the mean-square dis- 

 persion: 



1 d( X \) 



2 dt 



K 7 



Taylor (1921) showed that the mean-square dis- 

 persion of particles in a homogeneous random field 

 increases linearly with time, provided that the par- 

 ticles have been at large long enough for their present 

 motions to become statistically decorrelated from 

 their initial motions: 



PT. 



By definition then, the diffusivity associated with 

 homogeneous random motions is constant at /3/2, and 

 the dispersion of particles is governed by the advec- 

 tion-diffusion equation. Under these conditions the 

 diffusivity is synonymous with the "diffusion coeffi- 

 cient" of the Fickian advection-diffusion equation. 



Implications for the objective function 



The time-dependent nature of the mean-square dis- 

 persion has important implications for the nature of 

 the objective function used in estimating the param- 

 eters of the advection field. When the diffusivity is 

 constant, the positions of tags with initial conditions 

 i will follow a normal distribution with mean E[x T ] 

 and variance pT (Okubo, 1980). In addition, if the 

 recovery positions are reported with normally dis- 

 tributed errors, the observed positions of the tags will 

 follow a normal distribution with variance fiT+a 2 . 

 The maximum-likelihood estimates are therefore those 

 that minimize the weighted least-squares formula 



E (x iT - x iT ) 

 PT l+ a 2 ' 



(18) 



where x t is the predictor defined by Equation 8. In 

 effect, Equation 18 prevents tags that are expected 



to have large random displacements from dominat- 

 ing the analysis by down-weighting them according 

 to their time at large. 



The variance parameters in Equation 18, ji and 

 <7 2 , must either be known or estimated as part of the 

 search. However, if there is some evidence that the 

 observation errors are much larger than the displace- 

 ments attributable to random motions, then Equa- 

 tion 18 reduces to ordinary least squares. Similarly, 

 if the random displacements are much greater than 

 the observation errors, then Equation 18 reduces to 



{XlT ~ %ij* ' 



(19) 



There are many practical circumstances where it 

 may be reasonable to assume a constant diffusivity 

 and to apply Equation 18. Observations in the open 

 ocean suggest that the turbulent motions in many 

 regions are approximately homogeneous (e.g. de 

 Verdiere, 1983; Krauss and Boning, 1987; Figueroa 

 and Olson, 1989; Poulain and Niiler, 1989). More- 

 over, Porch ( 1993) points out that random walk mod- 

 els of fish movement also exhibit mean-square dis- 

 persions that increase in proportion to time (provided 

 swimming speed does not increase substantially dur- 

 ing the time period of interest). 



The maximum-likelihood formulation for an in- 

 homogeneous diffusion field is unclear. Equation 18 

 may be acceptable where the random displacements 

 of the tags increase monotonically with time, but it 

 will not be acceptable in all cases. When animals are 

 aggregating to spawn or feed, for example, the effec- 

 tive diffusivity would be zero or negative. Likewise, 

 the presence of coastal boundaries complicates the 

 matter because the diffusivity in one direction is zero. 

 For this reason, it may be more prudent to consider 

 methods that are robust to inhomogeneous variances, 

 such as least-median-of-squares regression (Rous- 

 seeuw, 1984) and least-absolute-value regression 

 (Bloomfield and Steiger, 1983). 



Estimating diffusivity 



As mentioned previously, the position of a tag at lib- 

 erty for time T in a homogeneous diffusion field fol- 

 lows the normal distribution with mean E[x (T ] and 

 variance pT. This implies that the displacement of 

 the tag relative to its expected position (D) is also 

 normally distributed with mean and variance flT. 

 The squared displacements are therefore gamma-dis- 

 tributed with parameters 1/2 and \K2(jiT+o 2 )). Ac- 

 cordingly, the maximum-likelihood estimates for ft 

 and a 2 are those that satisfy the constraints 



