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Fishery Bulletin 93(2), 1995 



gence to the true F values (Table 1 and Fig. 1, bot- 

 tom panel). 



True F 



Figure 1 



Progression in the approximation to the fishing mor- 

 tality rate for a catch equation not involving a plus 

 group (400 of the 1,000 pairs of approximated and true 

 F are shown). (Top) Initial approximation (Eq. 7); 

 (Middle) approximation after application of empirical 

 correction function (Eq. 13); (Bottom) approximation 

 after one iteration of Newton's Method (Eq. 12). 



Case II: with a plus group 



The initial approximations obtained for the plus 

 group problem were rather poor compared with those 

 of the catch equation without a plus group (Table 1, 

 Fig. 2, top panel). This was not unexpected, because 

 the plus group catch equation is not amenable to al- 

 gebraic manipulations that lead to analytical ap- 

 proximations. However, note that the initial approxi- 

 mation from Equation 11 was much better than that 

 from Equation 10: the observed A F I T F ratios indi- 

 cated smaller biases and a tighter approximation 

 overall (see Table 1). (Note: Subsequent statistics and 

 data reported in Table 1 and Figure 2 are based on 

 the approximation given by Equation 11.) 



In order to find empirical correction factors, we 

 plotted the observed A F I T F ratios against M (for dif- 

 ferent a values) and against a (for different M val- 

 ues) (see Fig. 3). Visual inspection of these figures 

 indicated that 1 ) the relationship between A F I T F and 

 M could be approximated by a linear model; 2) the 

 relationship between A F I T F and a could be approxi- 

 mated by a logarithmic model; and 3) there was an 

 interaction between M and a in terms of explaining 

 variability in A F I T F. Therefore, we fitted the 

 model 



A F/ T F = a + &! ln(a) + b 2 M + b 3 aM 



to the observed ratios, again by minimizing the sum 

 of absolute residuals. The empirical correction fac- 

 tor used was then 



emp - A F= inU - A FI{a Q +b l \a{a) 

 + b 2 M + b 3 aM) 



(14) 



with 



a = 0.9951, 

 b t = 0.2053, 

 b 2 = 0.0636, and 

 b 3 = 0.0161. 



This empirical correction function provided a sub- 

 stantial improvement in the approximations (Table 

 1, Fig. 2, middle panel). However, solution errors on 

 the order of 12% were still obtained after the correc- 

 tion. One iteration of Newton's Method was sufficient 

 to reduce the errors to within 2% (Table 1 and Fig. 2, 

 bottom panel) and the second iteration resulted in 

 virtual convergence (Table 1). 



