478 



Fishery Bulletin 98(3) 



of opaque zone completion based on the following obser- 

 vations. First, otoliths grow at a measurable rate that 

 allows both counts of annuli and measurements of otolith 

 weight to be converted to an estimate of age, with an 

 associated error ( Worthington et al., 1995). Second, this 

 is a continuous process best represented by a continuous 

 variable (sensu Zar 1996) — not a discrete variable such as 

 the (whole) number of opaque zones. Third, the widths of 

 outer increment cycles become similar for older fish (see 

 figures in Fowler, 1995). We therefore proposed that ratios 

 of otolith widths could be equated to ratios of corresponding 

 time intervals in a model if two approximations were 

 assumed: 



1 the rate of otolith growth throughout the formation of 

 an increment cycle was constant: and 



2 the widths of increment cycles on outer margins of 

 otoliths were the same. 



the measurements shown in Figure 3, these two values 

 were calculated as 



lP=^R,.x-T,>iiR,,,-R,) 



(1) 



where R, 



= the radius to the opaque zone formed 

 immediately before OTC injection: and 

 /?,^[ = the radius to the opaque zone formed 

 immediately after OTC injection; 



and as 



FF = (R-R)/(R-RJ. 



(2) 



where a = age of the fish; 



i?„ = the radius to the final opaque zone; and 

 ii!^,_j = the radius to the penultimate opaque zone. 



The terms R^^ and /?,^j were equivalent, and /?^_j and /?, 

 were equivalent, where there was only a single opaque 

 zone measured past the OTC mark. 



The cycle frequency (V) was estimated for all sections 

 with at least one opaque zone past the OTC mark as 



V = (1F + N + FF}/L, 



(3) 



where N 



the number of full increment cycles visible 

 outside the OTC mark; and 

 the time at liberty; the term TV = an integer 

 and N = in Figure 3. 



The "liberty fraction" (LF) was calculated as the sum of 

 the full cycles and partial cycles that formed during time 

 L, and was expressed in decimal fractions of cycles as 



LF = IF + N + FR 



(4) 



We obtained continuous variables by expressing mea- 

 surements of the marginal increment and distance between 

 the OTC mark and subsequent opaque zone as "fractions" 

 of widths of a completed increment cycle within an otolith. 

 The sum of these fractions and the counts of whole 

 cycles completed outside the OTC mark, divided by the 

 known time at liberty, produced estimates of the rate, or 

 periodicity, of opaque zone completion. Given this rate of 

 completion, and known dates of OTC marking and sacrifice, 

 the marginal increment could be used to estimate a date 

 on which the outermost opaque zone was completed. 



The number of increment cycles completed per year was 

 the "cycle frequency" (V^) and its inverse was the "cycle 

 period" in units of years. The "closing date" iCD) of an 

 increment cycle was defined as the date on which the 

 formation of the opaque zone was completed. The time 

 elapsed between OTC injection and the day on which the 

 fish was sacrificed was referred to as the "time at liberty" 

 (L) measured in decimal units of years. 



The "initial fraction" ^IF) and the "final fraction" (FF) of 

 otolith growth were the two ratios of widths modelled to 

 estimate time intervals with this direct method. Following 



The cycle frequency was investigated using 

 regression (SAS Institute Inc., 1989a) of the model 



LF = pL. 



(5) 



This assumed an intercept value of zero (there was no time for 

 otolith growth to occur when L=0) and the slope ^ = 1 when 

 cycles had an annual periodicity. Confidence intervals for [i 

 were calculated from t distributions ( Montgomeiy, 1991). 



The second model, to estimate closing dates iCD) of the 

 last cycle fully completed, assumed that for each otolith 

 section the cycles that finished during time L had equal 

 cycle periods. Closing dates were calculated as 



CD = K-365{FF/V), 



(6) 



where K = the date on which the fish was sacrificed, 

 coded by SAS in units of days (SAS Institute 

 Inc., 1989b). 



The calendar day of the year on which the cycle ended 

 (calendar closing date CCD) was derived from CD by 



