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Appendix 1 : An overview of the technical aspects of 

 the PFMC's rebuilding analysis 



The key steps of the PFMC's rebuilding analysis are 1) 

 to select the maximum allowable rebuilding time iT^^^), 



2) to develop specifications for projecting the population 

 size at the start of the current year-to-year T^^^, and 3) 

 to calculate the target fishing mortality rate so that the 

 probability of the spawning output rebuilding to 0.4B„ at 

 or before T^^,,^ equals a prespecified value, p^^.^, (taken to be 

 0.6 for purposes of the present study). 



Projecting the population forward and defining B^ 

 The population projections are based on the equation 



(A.l) 



where N^^ = the number of animals of age a at the start 

 of year y; 

 M = the instantaneous rate of natural mortality 



(assumed to be independent of age); 

 S^ = the selectivity for animals of age a; 

 F = the fully selected (i.e. S^^ ^ 1) fishing mortality; 

 R^, = the recruitment (both sexes) during year v; 

 "mm - ^^^ lowest age class considered in the model; 



and 

 •^max ~ '-^^ oldest age class considered in the model 

 (treated as a plus-group). 



The age structure of the population at the start of the 

 first year of the projection period is taken to be that from 

 the most recent assessment. A variety of approaches are 

 available to generate future recruitment (PFMC"). How- 

 ever, for consistency with the approach used in the bulk of 

 the rebuilding analyses conducted to date, future recruit- 

 ment is either based on randomly sampling recruitments 

 (with replacement) from a prespecified historical period or 

 based on randomly sampling the ratio of the recruitment to 

 the spawning output that spawned that recruitment (with 

 replacement) and then multiplying by current spawning 

 output. The choice between basing the projections on 

 sampling recruitments or sampling recruits-per-spawning 

 output is determined by regressing each of these on time 

 and selecting whichever has the lesser slope. The reason for 

 doing this is that the lack of a trend in recruits-per-spawn- 

 ing output is indicative of a stock-recruitment relationship 

 with low "steepness" (Francis, 1992), whereas the lack of 

 a trend in recruitment is indicative of a stock-recruitment 

 relationship with high "steepness." 



The pre-exploitation equilibrium spawning output used 

 to determine the rebuilding target is computed by mul- 

 tiplying the unfished spawning output-per-recruit by the 

 average recruitment over a prespecified number of histori- 

 cal years. Note that the range of years on which to base 

 the estimate of Bq will usually differ from that on which 

 generation of future recruitment is based. 



"> PFMC (Pacific Fishery Management Council). 2001. SSC 

 terms of reference for groundfish rebuilding analysis, 9 p. 

 Pacific Fishery Management Council, 7700 NE Ambassador 

 Place, Portland, OR 97220. 



