698 



Fishery Bulletin 93(4). 1995 



conventional tagging programs because the ability 

 to record the position of the tag is largely beyond the 

 control of the investigator. In this situation a suffi- 

 cient number of tags would need to be released in 

 each season and area strata to ensure that at least a 

 few would be recovered before straying into other 

 strata. The recovery times must also be short enough 

 to avoid biasing the results toward slower moving tags. 

 A more flexible estimation procedure, which is ca- 

 pable of incorporating trajectories that reflect the 

 combined effects of several different movement pat- 

 terns, can be derived by reformulating Equation 8 as 



t, n +T, 



J u{x,t) J 



dt. 



(10) 



If u(x,t) is held constant within specific space and 

 time strata but allowed to vary among strata, then 

 Equation 10 may be piece-wise integrated. The solu- 

 tion simplifies to 



^s+i ~t s ~ 



x s+l - 



when u > 0, and 



B n 



^s+l ~h~ 



when u < 0. 



l a+l,s 



x s+l ~ 



B a-\- B a 



u a-\,s 



U n,s +B ii' 



(11) 



u n,s + ^n+i> 



(12) 



Here u as = velocity in area a during season s; 

 B a = boundaries of the areas (see Fig. 1); 

 a = first area occupied by the tag during 



season s; 

 £2 = last area occupied by the tag during 



season s; 

 t s = date at the onset of the s'th season; and 

 x s = position of tag at onset of s'th season. 



With these recursions, the position of the tag at 

 the end of each season can be computed from the 

 tag's position at the end of the previous season by 

 using an estimate of the advection field. This proce- 

 dure can then be applied sequentially to compute the 

 expected position of the tag at the date when it was 

 recovered from its initial position. The "best" esti- 

 mates of the strata-specific advection rates would 

 minimize some objective function of the differences 



Area a Area a+1 



Area a+2 



► ► ► 



Sa+2 



Ba*3 



Figure 1 



Schematic of a region divided into three areas illustrating 

 the definition of the area boundaries, B. 



between the observed recovery positions and those pre- 

 dicted with the recursions. (The appropriate objective 

 function will be discussed in the section on diffusion. ) 

 The sequential procedure itself is accomplished by 

 first determining the starting season s and starting 

 area a for each tag from the date and position where 

 it was released. The position of the tag at the end of 

 the first season (start of season s +l) can then be 

 obtained from the recursions, by replacing x and t 

 with the position and date of release. The recursions 

 are then applied as given until the last season, which 

 is determined by the recovery date. Finally, the formula 

 for estimating the recovery position is obtained by re- 

 placing t s+l in the recursions with the recovery date. 



Continuous models 



The approach proposed in this section involves de- 

 veloping an adequate continuous model of the ad- 

 vection field, u(x,t), and solving Equation 8 for* as a 

 function of t. To illustrate, consider a fish population 

 migrating out of a basin. Suppose each fish swims 

 initially at speed u Q in the positive x direction and 

 increases its speed as it proceeds. Further, suppose 

 that periodically the fish are either helped or hin- 

 dered by sinusoidal oscillations in the water currents. 

 A reasonable model for the velocity of the fish might 

 be u(x,t) - U(.+ ax + bsin[ct]. The solution is 



._ "o 



, asin[c£] + ccos[c^] at 

 o— — s 7. (-/e 



a 2 +c 2 



(13) 



where 



| u Q | 6 asin[c< ] + ccos[c< o n g , n 



a 2 +c 2 



The best estimates of the parameters u ,a,b, and c 

 would be those that minimize some appropriate func- 



