Porch: Estimating velocity and diffusion from tagging data 



701 



T i lDf-pT l -a £ )_ 



^2n2 



and 



X^ 



Df-jiT—o 2 



f (pT,+6 2 ) 2 



= 



0, 



where Z), = x iT - x iT . These equations must be solved 

 numerically. When o 2 is negligible, however, the 

 maximum likelihood estimator for (i reduces to 



iv# 



Htt 



n- T, 



Less efficient estimates of fi and o~ 2 can be obtained 

 from a linear regression of the squared displacements 

 on the time at liberty: 



D? =mT l +b + e l , 



where e is a random deviation term. The slope of the 

 regression m is an estimate of fi and the intercept b 

 is an estimate of a 2 . 



Estimator performance 



This section presents the results of a series of sto- 

 chastic simulations designed to test the efficacy of 

 trajectory-based estimators. Although no attempt 

 was made to exhaust the possibilities, a sufficiently 

 broad range of conditions was examined to validate 

 the methodology. The experimental design and re- 

 sults are discussed in the subsections below. 



Experimental design 



A factorial design was used to study the behavior of 

 the estimator in response to the advection field, level 

 of diffusivity, distribution of recovery rates, and num- 

 ber of tags recovered. Two hundred test data sets 

 were generated for each possible combination of fac- 

 tors. The estimators were then applied to each of the 

 test data sets. 



Response variables The accuracy of the estimators 

 was quantified by the percent error of the parameter 

 estimates ( 6) relative to the actual values (0 t ): 



200 



6 , - 0,, 



200 £ t true 



100%. 



The precision of the estimators was quantified by the 

 coefficient of error (CE), 



CE = 1007c 





6>,-0„ 



7=1! 



0* 



The coefficient of error is analogous to the familiar 

 coefficient of variation except that the true values of 

 the parameters are used in place of the average of 

 their estimates. Alow CE, therefore, implies that the 

 estimates are both unbiased and precise. 



Factor levels Two advection models were consid- 

 ered: the sinusoidal model dx/dt = u + ax + bsin[ct], 

 discussed earlier in connection with Equation 13, and 

 a discrete model with two areas and semi-annual 

 seasons. The parameters of the first model — w , a, b, 

 andc — were valued at 8.6 kmday -1 , 0.004 day -1 , 17.3 

 kmday -1 , and 2;t/365 day -1 , respectively. The para- 

 meters of the second model are the area and season- 

 specific constant advection rates. The rates in areas 

 1 and 2 were set equal to 5 and 10 kmday" 1 during 

 the first season and -7 and -4 km- day * during the 

 second season. Area 1 extended from negative infin- 

 ity to 1,000 km and area 2 extended from 1,000 km 

 to positive infinity. 



Two diffusivity (fil2) levels, 0.95 and 822 km 2 -day" 1 , 

 were examined. These levels were derived by assum- 

 ing that tagged fish move according to the bilateral 

 random walk model (see Porch, 1993) with an aver- 

 age speed of 0.5 or 6 meters per second and change 

 direction an average of once per minute or once ev- 

 ery five minutes, respectively. 



The effects of variations in tag recovery rates were 

 evaluated by dividing the relevant spatial domain 

 into two zones and by varying the likelihood of re- 

 covering a tag between them. Three such scenarios 

 were considered. In the first, the probability of re- 

 covery was the same in both zones. In the second, 

 the probability of recovery was ten times higher in 

 zone A than in zone B (1.0 versus 0.1). In the third 

 scenario, no tags were recovered in zone B. The 

 boundary separating recovery zones A and B differed 

 with the advection models. The demarcation point 

 was x - 400 km in the sinusoidal model and x = km 

 in the discrete model. 



Test data Each test data set was generated by simu- 

 lating the individual paths of a prescribed number 

 of tags (n). The release positions were randomly as- 

 signed values between and 200 km when the sinu- 

 soidal advection model was used and were between 

 -1,000 and 1,000 km when the discrete model was 



