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Fishery Bulletin 95(2), 1 997 
ery is to correct the catch values to reflect the as- 
sumption of a constant fishing mortality rate (F). The 
catch matrix for VPA would no longer contain the 
observed numbers of fish caught, but rather the num- 
bers of fish that would have been caught under the 
assumption of an annual F. This paper presents a 
simple method for this conversion along with ex- 
amples of the reduction of bias due to the method 
and a discussion of further applications. The Fortran 
source code and the executable program for this cor- 
rection process are available from the authors. 
Methods 
The algorithm for correcting annual catches from 
seasonal fisheries to meet the assumption of a constant 
fishing mortality rate during the year is as follows: 
Let i = 1, 2, . . ., K index time intervals (not neces- 
sarily of equal length) during the year; 
A /. = the length of time in years for interval i; 
M = annual natural mortality rate; 
C ( = observed catch in numbers during interval 
i; 
N' = population numbers at the start of inter- 
val i) 
F = fishing mortality rate during interval i\ and 
F a = annual fishing mortality rate. 
For each year, age cell in the VPA catch matrix: 
1 Assume a value for N K+V 
2 For each time interval progressing backwards 
from if to 1. 
2a Solve for F given C-, N i+V M, and A t i from the 
catch equation: 
C. 
N l+1 e MAt ' +F ' F t (l- e~ MAI ~ F ‘) 
MM, + F 
2b Compute AT. given N i+V M, AC and F from the 
exponential decline equation: 
N t =N l+1 e l 
3 Compute F A that reduces N 1 to N K+1 given M as 
F a =-M- In 
N k+ i 
N, 
4 Compute annual catch (C A ) under F A , given N x 
and M from catch equation: 
N 1 F A (l-e~ M ~ F ' A ) 
A M + F a 
C A is the corrected catch to be used in VPA for the 
given year and age. Once all years and ages in the 
catch matrix have been corrected, the population 
abundance matrix can be estimated through virtual 
population analysis with the corrected catch values 
in place of the observed catches. The resulting popu- 
lation abundances at the end of the year can be used 
as the assumed values for N K+1 in step 1 and the 
process repeated to generate a recorrected catch 
matrix. Note that the observed catches are still used 
in step 2a of the algorithm; it is only the N K+1 values 
that change from computing the corrected to com- 
puting the recorrected catch. The recorrected catch 
matrix can again be used in virtual population analy- 
sis to estimate the population abundance matrix, and 
this iterative procedure can be repeated until the 
corrected catches do not change value. 
This iterative process will produce population num- 
bers from virtual population analysis that are con- 
sistent with the assumption of a constant fishing 
mortality rate during the year. Each annual catch in 
the VPA matrix is treated individually for the cor- 
rection and then all the corrected catches used as 
input for VPA. The purpose for the iteration is to give 
a more solid basis for the choice of N K+] for each ob- 
served catch (step 1 in algorithm) because the cor- 
rected catch value depends on the choice ofiV A+1 (Fig. 
2). Guessing too high a value for N K+l results in an 
underestimation of the corrected catch and vice versa, 
although, in general, the magnitude of bias is less 
for choosing N K+1 too large than too small. The tim- 
ing of the catch also impacts the amount of bias in 
the corrected catch; earlier catches are slightly less 
biased than later catches (Fig. 2). The high biases 
found with low guesses for N K+1 correspond to ex- 
tremely high values of the fishing mortality rate (Fig. 
3, top panel) and are due to the catch removing a 
large portion of the population (>90%). When the 
catch is not removing such a large proportion of the 
population, a wide range of guesses for N K+1 will re- 
sult in similar corrected catches (Fig. 3, bottom 
panel). The direction of the change between observed 
and corrected catch depends more upon the time of 
the catch than the N K+1 though (Fig. 4). It should be 
noted that the apparent linear relationship between 
corrected catch and time of the catch shown in Fig- 
ure 4 is due to the values of N K+1 and M used in the 
example and will not always occur. The use of VPA 
results for values of N K] ensures a reasonable cor- 
rected catch value. 
Once a value of N K+1 is chosen for an age cell of a 
given year in the VPA catch matrix, either from a 
