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Fishery Bulletin 100(3) 



Uncertainty 



Uncertainty in estimates of F^jgy is likely larger than 

 typically measured by variance estimates in assessment 

 models because uncertainties in catch, the assumed natu- 

 ral mortality rate, somatic growth, and other factors are 

 generally not included in stock assessment model variance 

 calculations. In simulation runs including uncertainty, we 

 used 



 MSY. s 



■V + f , 



(3) 



where f^/i;y, was used in simulation s, Fyi,y was the "best" 

 estimate, and e^ was drawn from a normal distribution 

 with mean zero and variance a'j = CV^Fj^^^.y.The CV (20%) 

 assumed in our simulation runs implies that the true F^^jgy 

 is within ±40*^^^ of the best estimate with about 95% prob- 

 ability. We truncated Fi^^'iv values at ±50% F^^gy to avoid 

 implausibly small (including negative) or large F^^gy^ 

 values. These ad hoc bounds seemed reasonable because 

 they were slightly larger than the 95% confidence interval 

 implied by the CV for uncertainty (±40%). 



Our assumptions about uncertainty in F^gy are crude 

 but seem reasonable based on our experience and by 

 analogy to uncertainty about natural mortality rates (M), 

 which are sometimes used as a proxy for F^jgy (Clark, 

 1991). In assessment work, a stock with an assumed 

 natural mortality rate M=0.2/yr, for example, might have 

 a "subjective" uncertainty range of about ±40% (i.e 0.12- 

 0.28/vr). It seems reasonable to assume that uncertainty 

 about M and F^/gy would be similar 



Process errors 



We modeled process errors as potentially autocorrelated 

 random changes in the intrinsic population growth 

 parameter (r^,). Previous analyses used independent or 

 autocorrelated random errors in realized annual produc- 

 tion rates idB^/dy. Sissenwine, 1977; Gleit, 1978; Shep- 

 herd and Horwood, 1980; Ludwig, 1981; Sissenwine et 

 al., 1988), or independent random errors in recruitment 

 (e.g. Getz, 1984), next year's biomass (B,^,, Ludwig et al.. 

 1988), or surplus production (Doubleday, 1976). Produc- 

 tion process errors may be independent in some cases but 

 were autocorrelated for both of our example stocks (see 

 "Results" section). 



Our analysis, like Sissenwine et al.'s ( 1988 ), includes au- 

 tocorrelated errors because they affected rebuilding times 

 in preliminary model runs, are biologically plausible and 

 widely recognized (favorable and unfavorable conditions 

 for production seem to persist for more than one year in 

 many stocks), and because correlated errors were obvious 

 in production model fits for Georges Bank yellowtail floun- 

 der and cowcod rockfish. In contrast to previous studies, 

 we estimated variances and autocorrelations for stochas- 

 tic r^, values in our simulation models from available data. 

 In addition, our simulation models used lower bounds for 

 r^, based on the natural mortality rate. 



We used the gamma distribution (Johnson et al., 1994, 

 Appendix 1) to describe process errors in the produc- 



tion model because it is flexible, asymmetrical (like our 

 estimates of production process errors for cowcod rock- 

 fish), and (in the three-parameter form) accommodates 

 negative r\, values. We devised a simple way to simulate 

 autocorrelated process errors from a distribution nearly 

 identical to a gamma distribution used to simulate uncor- 

 related process errors. This makes comparisons between 

 runs with and without autocorrelation easier. Sissenwine 

 et al. (1988) also used a gamma distribution for produc- 

 tion process errors because simulated state variables in 

 logistic models (with constant catch and Gaussian process 

 errors on the realized production rate dB^.ldy) have distri- 

 butions that resemble a gamma distribution (Dennis and 

 Patil, 1984). 



The first step in modeling production process errors was 

 to obtain empirical estimates of variance and autocorrela- 

 tion. Based on stock assessment results, surplus produc- 

 tion in each year (P^) was computed with the following 

 equation: 



fiv.l-fiv+C, 



(4) 



where C^ = catch data was catch; and 



B^ = estimated biomass at the beginning of year y. 



The discrete time version of our logistic model with pro- 

 cess errors is 



n = 'vfiJi 



K 



Solving for i\, gives 



P,.K 



BaK-B, 



(5) 



(6) 



where B^ should be no larger than, say, 95% K to avoid 

 unrealistic values of 7\, that are calculated when positive 

 production is observed in stock assessment results at bio- 

 mass levels near or above estimates oi K. As shown in the 

 "Results" section, empirical estimates of variance a- and 

 autocorrelation (p) for i\. values were relatively insensi- 

 tive to assumptions about A'. The variance of observed 

 i\, values includes both process and measurement errors 

 and is an upper bound estimate for the variance due to 

 process errors only. In other words, results of our simula- 

 tion analyses may overstate the importance of production 

 process errors in rebuilding overfished stocks because our 

 variance estimates may be too large. 



We used -M (where M is the instantaneous natural 

 mortality rate assumed in the stock assessment) as a 

 lower bound on j\ in simulations. Negative r^ values are 

 common in some stocks (e.g. 36% and 17% of years for 

 anchovies [Engraulis spp.] and sardines [Sardinops and 

 Sardina spp.], Jacobson et al., 2001) because stocks can 

 decrease in biomass from one year to next with no fishing 

 and because negative values are occasionally seen in real 

 data sets (e.g. Myers et al., 1999). If process errors are ig- 

 nored and M is constant, then 



