FISHERY BULLETIN: VOL. 75, NO. 3 



and 



log, a* '- a x (2A' k - A' k -i - A* + 1 ) 



+ a 2 (2Ak 2 - A£ x - A* 2 +1 ) 

 + a 3 (2A' k 3 - Af-i -AkW. 



Following Equations (4.1) to (5.1) steps, we get 

 an expression for q~k similar to Equation ( 5 . 1 ) with 

 4>A' as 



M'= a 2 (2Af - Ak 2 -! - Ai 2 +1 ) 



+ a 3 (2A' k 3 - Ak 3 -i - A£x ). (5.9) 



ESTIMATION OF CATCHABILITY 



Denoting all terms of the numerators of Equa- 

 tions (5.1), (5.2), and (5.3) with the exception of 

 \og e (Clf)'k by "p" and their denominator by 4>F; 

 the equations become 



(Jk 



\o gi AC/fV h + p 



<t>f 



(6.1 



Equating p to zero, a first estimate for q k is ob- 

 tained which is used together with catch data to 

 estimate A', R k , M k , and 4>A' so that p can be 

 estimated and used to obtain the required esti- 

 mate for q k as well as other parameters. 



If p has a negative sign, this means that the 

 first estimate for q^ was higher than the true value 

 and p/4>f is the correction to be subtracted to ob- 

 tain the improved estimate and the reverse holds 

 good as will be shown by the solved example. 

 Equation (6.1) is therefore betterly transformed to 



Qk 



<t>f 



+ 



0/ 



(6.2) 



Solved examples showed that one single correc- 

 tion is sufficient to obtain precise estimates for 



q k for populations with increasing or decreasing 

 abundance which is a great advantage. 



If a number of equations like (6.2) are available, 

 they may be combined in a single expression as 



Qk 



^ log,(C//-U 



+ 



Zp 



!<*>/■ 



(6.3) 



EXAMPLE 



Detailed informations are required to use the 

 equations given above for estimating correctly the 

 catchability as dividing sampling surveys into 

 groups coinciding with seasons having more or 

 less constant population parameters like periods 

 with high, low, or nil recruitment, migration, 

 natural mortality, and catchability. 



As published data reviewed by the author 

 lacked such information, it was decided to treat 

 the hypothetical example given by Rafail (1974) 

 so as to demonstrate the advantage of the above 

 modified equations. Table 2 shows a part of 1974 

 example containing periods I and III with increas- 

 ing and decreasing abundance, respectively. 



Computations for Period I 



A) Surveys 1, 2, and 3 



\og e (Clf)' k = lo&( 1.001 18) = 0.00116 

 (bf= 0.5(1,000-2,000) = -500 

 q k = -0.00116/-500 = 2.320 x 10 K . 



Above ^-estimate is used to evaluate A', (Rk - 

 M k ), and <t>A ' using the relations: 



F' k --= q k fk,N k0 --= catch/F^ 

 exp{Ap = N k + 1 /N k 

 Rk-Mk= Ai + FJ 



A'=Ri- Mk xx - Fi . 



TABLE 2. — A hypothetical example showing sampling periods I and III with increasing and 



decreasing abundance. 



566 



