Restrepo et al. Monte Carlo simulation applied to Xiphias gladius and Cadus morhua 



737 



known. For example, similar trends in population abun- 

 dance may be obtained when two very different values 

 of natural mortality rate are assumed as inputs; despite 

 the simOar trends (and hence correlation with the abun- 

 dance indices), there may be large differences in the 

 absolute estimates of abundance. 



In this paper we use the term "uncertainty" to 

 describe any variability or error that arises during the 

 stock assessment process. Uncertainty can enter into 

 an assessment in various ways. There may be uncer- 

 tainties in the values of the inputs, e.g., the total catch 

 may be estimated with error. Also, the formulation of 

 the assessment model may be subject to uncertainty, 

 and the analyst may make data-dependent decisions 

 during the analysis which are subject to error. The 

 degree to which these sources of error are incorporated 

 into the analyses will determine the perceived uncer- 

 tainty in the overall assessment results. If all sources 

 of error are not appropriately accounted for, then 

 estimates of the uncertainty in the assessment results 

 may be too small. 



Monte Carlo simulation is a convenient tool for study- 

 ing a model's outputs given different types and levels 

 of error in the model's inputs (e.g., Restrepo and Fox 

 1988). In a sensitivity analysis framework. Pope and 

 Gray (1983) and Rivard (1983) used a Monte Carlo 

 approach to study the relative contribution of various 

 inputs to the overall uncertainty in total allowable 

 catch (TAC) estimates obtained from calibrated SPAs. 

 Francis (1991) used Monte Carlo simulation analysis 

 to construct risk curves describing the chances of not 

 meeting management objectives as a function of the 

 catch quota. In this paper, we present a general 

 method, also based on Monte Carlo simulation, to ac- 

 count for uncertainty in assessment results, including 

 the parameters directly estimated from the SPAs as 

 well as derived statistics used to set management 

 targets and allowable catches. We also show how the 

 simulation results can be used to quantify the risk (of 

 not meeting a management goal) associated with the 

 selection of a given TAC, and we describe a measure 

 of the cost of picking a conservative catch quota. 



We apply the simulation method to swordfish 

 Xiphias gladius in the North Atlantic Ocean and cod 

 Gadus morhua off eastern Newfoundland and 

 southeastern Labrador. These fisheries are quite dif- 

 ferent in nature. Swordfish are highly migratory, 

 managed internationally with fishing mortality con- 

 trols, and the data set allows for the estimation of 

 only a few parameters in models with many con- 

 straints. Northern cod, on the other hand, are demer- 

 sal, managed by quota, and the availability of age- 

 specific survey indices allows for the estimation 

 of many parameters with a minimum number of 

 assumptions. 



Quantifying uncertainty 

 by simulation 



Suppose the only uncertainty in the inputs to an assess- 

 ment model concerns the value of the instantaneous 

 natural mortality rate, M, and that M could be 

 anywhere in the interval 0.15-0.25/yr with equal 

 likelihood. One could compute the assessment model 

 results for a large number of uniformly spaced values 

 of M in this interval (e.g., 100) and make histograms 

 of the results. This would represent the perceived in- 

 formation about the relative likelihood of the estimated 

 output taking on various values. If not all values of M 

 were believed to be equally likely, one could weight the 

 100 outputs by the probability associated with the cor- 

 responding inputs. 



The above procedure becomes awkward when there 

 are a number of inputs subject to uncertainty, because 

 the number of combinations of input parameter values 

 becomes very large. An alternative is to use a Monte 

 Carlo approach in which values of the inputs are drawn 

 randomly from probability distributions. A sufficient- 

 ly large number of plausible input data sets are thus 

 generated and used to compute the assessment model 

 results such that the distributions of the estimated out- 

 puts are clearly defined. This may involve hundreds or 

 thousands of runs, depending on the types of data and 

 models used (in our work we found that 500-1000 data 

 sets were necessary to obtain stable results). 



A typical assessment of a fish stock using SPA in- 

 volves three levels of analysis. First, data are prepared 

 for the SPA. This usually involves estimating and age- 

 ing the annual catch, and computing indices of abun- 

 dance for calibration. Second, the SPA itself is carried 

 out (it is also frequently termed "VPA," for Virtual 

 Population Analysis). In many cases, several SPAs are 

 carried out to examine the goodness-of-fits of the in- 

 put data to alternative model formulations or simply 

 to examine the sensitivity of the results to the alter- 

 native formulations. Third, derived statistics are com- 

 puted. These are commonly biological reference points 

 (Fmax, Fq i; Gulland and Boerema 1973), and forward 

 projections of stock status and catches under alter- 

 native management actions. 



It is easy to see how the Monte Carlo approach 

 can be used to characterize the uncertainty in the 

 entire analysis process, starting with the raw data 

 collected for the first step in the above procedure. 

 For instance, the total annual catches and their pro- 

 portions at age can be obtained by resampling the 

 original data that led to the catch estimates, through 

 a non-parametric bootstrap (Efron 1982). These boot- 

 strapped catches would then be used in the SPAs, 

 whose results, in turn, would affect the values of 

 projected future catches. 



