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Fishery Bulletin 93(4). 1995 



employed. The release dates were randomly assigned 

 values between day and day 365. These choices 

 mimic an opportunistic tagging program where the 

 number of tags released is relatively constant dur- 

 ing the year. The recovery dates were obtained by 

 randomly selecting the liberty times from an expo- 

 nential distribution with parameter Z (instantaneous 

 mortality rate) equal to 0.4 yr -1 . 



The displacement of each tag due to advection was 

 computed from its release position and from release 

 and recovery dates by using the solutions to the de- 

 terminate advection models described above. The 

 solution to the sinusoidal model is given by Equa- 

 tion 13, and the solution to the discrete model is given 

 by Equations 11 and 12. The diffusive effect of ran- 

 dom motions was then simulated by adding a ran- 

 dom normal deviate with mean and variance 

 pT km 2 . Next, an acceptance-rejection criterion was 

 invoked to determine whether or not the tag would 

 be recovered. Candidate tags located in recovery zone 

 A were unconditionally accepted, but candidate tags 

 located in zone B were accepted only with probabil- 

 ity P ( = 1.0, 0.1, or 0). This was done by generating a 

 uniform random number between and 1 and by 

 excluding the tag if that number was greater than 

 the prescribed recovery probability. 



The process described above was repeated until a 

 total of n recovery positions were accepted. Normally 

 distributed errors (with variance 25 km 2 ) were then 

 added to each of the accepted release and recovery 

 positions to simulate imprecise position reporting. 

 In this way an artificial sample of n tag recoveries 

 was created. 



Estimation The predictors were fitted to the test 

 data by minimizing the weighted least-squares sur- 

 face described by Equation 19. The predicted posi- 

 tions were calculated by substituting estimates of the 

 parameters into the same advection equations used 

 to generate the data. This allowed the analyses to 

 focus on the interactions of recovery rates and veloc- 

 ity variance without the confounding effects of model 

 misspecification. 



The minimization was accomplished by using the 

 Nelder-Mead simplex algorithm AMOEBA (Press et al., 

 1986), which, although slower than derivative-based 

 methods such as Marquardt's algorithm, is less sensi- 

 tive to the discontinuities in the solution surface asso- 

 ciated with discrete advection models. Heavy penal- 

 ties were imposed to prevent the search from extend- 

 ing beyond the bounds of a reasonable domain. For 

 example, the maximum possible sustained speed of a 

 migrating tuna might be the sum of the cruising speed 

 of the fish and the maximum speed of the water cur- 

 rents. 



The AMOEBA search was restarted at the point 

 P Q , where a minimum had been found, to avoid local 

 anomalies in the solution surface. Subsequent "re- 

 starts" continued until five consecutive sets of pa- 

 rameter estimates differed by less than one percent. 

 New vertices were selected for each restart by using 

 the formula 



Pu = P 



0j c 



,0.5A5, 



(i,j = X 



,co) 



where P. is the value of the/th coordinate (param- 

 eter) in the z'th vertex of the initial simplex, K is a 

 standard normal variate, and 8- is equal to one if i 

 equals j and zero otherwise. 



Results 



This section is divided into two parts, each focusing 

 on the results pertaining to one of the two types of 

 advection models. 



Sinusoidal model 



The estimation procedure generally behaved very 

 well when the frequency (c) of the sinusoidal oscilla- 

 tions was known and the diffusivity was low (0.95 

 km 2 -day _1 ). The CE's, which reflect both accuracy and 

 precision, were very low regardless of the distribu- 

 tion of recovery rates (Fig. 2). For the most part, the 

 estimator continued to perform well at high 

 diffusivities ( 822 knrday 1 ). When the recovery prob- 

 abilities were the same in both zones, the estimates 

 were unbiased and the CE's rapidly decreased with 

 increasing sample size to less than 10 percent. The 

 estimates were only slightly biased and similarly 

 precise even with a tenfold difference between the 

 recovery probabilities. It was only when no tags were 

 recovered in zone B (i.e. beyond the 400-km demar- 

 cation) that the estimates were significantly biased. 

 The trends were very similar to those described 

 above when the frequency parameter c was estimated 

 along with the other three parameters. A few of the 

 runs, however, failed to converge to acceptable solu- 

 tions — the weighted least-squares function being an 

 order of magnitude greater than that expected, given 

 the known diffusivity. This problem is not surpris- 

 ing considering the oscillatory nature of any peri- 

 odic surface. Even if the true values of the other pa- 

 rameters were known, the surface map of the objec- 

 tive function would be characterized by local peaks 

 and valleys that vary with the estimate of c. This 

 behavior is demonstrated by a simplified model of the 

 residual sum of squares (Fig. 3). Although the ampli- 

 tudes of the peaks and valleys in the more complicated 



