468 



Fishery Bulletin 89(3). 1991 



Discussion 



The method presented is a variation on previous tech- 

 niques of obtaining age estimates from microincrement 

 densities (Methot 1983, Ralston and Williams 1988a, 

 Gauldie et al. 1989). It provides a direct estimate of 

 the fit of the rate function used to approximate the 

 otolith growth curve, which is then integrated essen- 

 tially without error. Part of the difficulty in compar- 

 ing the present age estimates with those of Ralston and 

 Williams was that a direct measure of error was not 

 available for their estimates. Within the constraints of 

 the fit of each rate curve, the present technique pro- 

 vides an accurate estimate of age at any chosen inter- 

 val of otolith radius. The method also allows age esti- 

 mation from the otolith focus to its perimeter, where 

 readings become difficult, and gives a reliable estimate 

 of the time transpired between any two points on the 

 otolith radius. 



The error introduced in numerical integration by the 

 trapezoid method, dividing the otolith radius into in- 

 crements of equal length and using the mean growth 

 rate in each interval to estimate age, is well understood 

 (Salas et al. 1986, Atkinson 1989). Most alternative 

 methods of numerical integration have been developed 

 to reduce the error introduced by the trapezoid esti- 

 mate. Romberg's method is one of the most effective 

 ways of eliminating this error, evaluating and sub- 

 tracting it out as part of the process of integration. The 

 result is a negligible integration error for the purposes 

 of this discussion. The error in the present method 

 derives from natural variability in otolith growth rates, 

 lack of fit of the chosen rate curve, and any inade- 



quacies in the data. The first two sources of error are 

 measured by the sums of squares, and are represented 

 in the multiple correlation coefficient and asymptotic 

 confidence intervals for parameter estimates. Thus, 

 another advantage of the present method is that alter- 

 native curves can be fit to evaluate the advisability of 

 using other functions or combinations of functions to 

 fit the data. 



The reliability of these growth estimates is depen- 

 dent, among other things, upon the assumption that 

 Ralston and Williams were able to count all or most 

 of the increments present in the otoliths at a given 

 radius. This is true to a varying extent; however, the 

 reliability of the readings may decrease as fish get 

 older. Most microincrement density measurements 

 were made within 50-60% of the total otolith radius. 

 However, in each region, there were "legible" seg- 

 ments near the perimeter of some otoliths. The section- 

 ing techniques and optical resolution of the Nikon light 

 microscope and RCA video system used to observe the 

 slides were sufficient to see detail considerably below 

 the scale on which microincrements were recorded. 

 Therefore, we believe that any error in the data is 

 unlikely to have been derived from a failure to observe 

 increments that were present. 



Some error in measuring total otolith radius may 

 have occurred, however, due to changes in the orien- 

 tation of growth along the postrostral axis as fish get 

 larger. Difficulties in reading otolith microincrements 

 in older fish, induced by the transformation in shape 

 and crystalline structure of the otolith affect the 

 legibility of daily increments (Radtke 1987, Davies 

 et al. 1988). Thus an important follow-up to these 

 preliminary estimates is to confirm the orientation of 

 the postrostral growth axis as a function of otolith size 

 for each region. The hypothesis that otolith growth rate 

 is essentially constant beyond a certain radius could be 

 tested by reexamining microincrement densities for 

 some of the larger otoliths near their perimeter. 



A difference in actual microincrement densities at 

 maximum otolith radius would influence age estimates 

 significantly. An unmodified Gompertz rate curve 

 (without the constant) was used in early attempts at 

 fitting microincrement densities. This curve fit the data 

 fairly well for small otolith radius, but the decline of 

 the rate curve to zero at greater radius provided a poor 

 fit and caused age estimates for large fish to increase 

 by more than an order of magnitude. A constant was 

 added to the rate curve for these reasons, significant- 

 ly improving the fit and bringing age estimates into 

 conformity with other estimates by fishery biologists 

 and experienced fishers. 



The growth estimates presented can be improved by 

 sampling a wider size range and larger number of 

 organisms and by recording microincrement densities 



