FISHERY BULLETIN: VOL. 75, NO. 3 



</,A' = 0.04(0.0037080-0.0074535) 

 = -0.00015 

 (f)A'/\(j)f\ = -0.00015/15,000= -0.00001/1,000 

 = -0.01 x 10" 6 

 q k = (2.008-0.010)10 6 = 1.998 x 10" 6 . 



Using the more accurate A± -estimates we get 



(l)A' = 0.04(0.00673496-0.01108147) 

 = -0.000174 

 <t>A'/\4>f\= -0.000174/15,000= -0.011 x 10" 6 

 q k = (2.008-0.01D10- 6 = 1.997 x 10" 6 . 



Using Equation (5.8) and the more accurate 

 (R' k - M/;)-estimates, we get a similar result as 



<t>A' = -0.08< -0.01787)1 -0.02008) 

 + 0.04( -0.0036289) 

 = -0.0000287-0.0001451 = -0.000174. 



The above example shows that the so-called less 

 accurate estimates gave equivalent results to the 

 more accurate estimates. However, in situations 

 with variable (R' k M' k (-values it will be pref- 

 erable to compare their results with those to be 

 obtained with the more accurate values. 



DISCUSSION 



Rafail (1974) showed the great advantages of 

 his method for the estimation of some important 

 parameters of fish populations like catchability, 

 fishing mortality, natural mortality, and recruit- 

 ment from catch data. He also showed that a 

 similar analysis of data of tagged fish can allow 

 the estimation of other important parameters like 

 migrations and at the same time may correct the 

 estimates of parameters of untagged fish that 

 may be biased by unexpected recruitments and 

 migrations. 



The modifications presented here for expres- 

 sions used to estimate catchability cause a great 

 simplification, shortening of calculations and 

 more accurate results. Rafail (1974) gave in his 

 table 4 a summary of results of HP-20 computer 

 programme for iteration of period I with increas- 

 ing abundance. The results of the computer 

 showed that after 16 iterations with a precision 

 at six decimals and 22 iterations with a precision 

 at nine decimals; q was estimated at 1.92 x 10 -6 

 and 1.83 x 10 6 , respectively. The corresponding 

 estimate by the present modified expressions was 

 1.824 x 10 6 by a single step. This simplification 



allowed the estimation of q from the next series of 

 sampling surveys of period I (2, 3, and 4) so that 

 an overall estimate of 2.003 x 10 6 becomes avail- 

 able which is highly accurate as the original value 

 is 2 x 10' 6 . 



As far as period III with decreasing abundance 

 is concerned, we find that 1974-expressions gave 

 after three iterations 1.98 x 10 6 while the new 

 expressions gave after one step 1.998 x 10 -6 or 

 1.997 x 10" 6 for q compared with an original 

 value of 2 x 10~ 6 . 



It is, therefore, concluded that the present modi- 

 fied expressions allow better accuracy and large 

 economy in calculations and time during estimat- 

 ing q as compared with 1974-expressions. This 

 greater accuracy of q will allow better estimates 

 for other parameters. It appears what is a logic 

 conclusion that the larger number of surveys, the 

 larger will be the number of available g-estimates 

 allowing a more accurate evaluation for catch- 

 ability and other parameters. 



SUMMARY 



Modifications are presented here for expres- 

 sions given by Rafail (1974) for estimating catch- 

 ability to evaluate fishing and natural mortalities, 

 recruitment, and migration assuming seasonal 

 and subpopulation variability and the constancy 

 of the parameters within the seasons. These modi- 

 fications depend on the relation 



(exp(A^) - l)IA' k = exp(±0.5A* + 0.04A* 2 ) 



where A' k denotes the instantaneous rate of 

 change offish abundance during the kth sampling 

 period. The above expression is an extension of 

 Paloheimo (1961) expression and gave a maxi- 

 mum error less than 1% when A ' lies between 

 ±3.0 and smaller errors at smaller values of A 

 so that the errors are less than 0.2^ when A' 

 lies between ±2.5. This expression can be consid- 

 ered as highly accurate in the range that is always 

 encountered in fisheries studies. 



The modified expressions allow a large economy 

 in calculations and time and a better accuracy 

 for the estimation of catchability. 



LITERATURE CITED 



ALLEN, K. R. 



1966. Some methods for estimating exploited populations. 

 J. Fish. Res. Board Can. 23:1553-1574. 



568 



