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Fishery Bulletin 92(4). 1994 



where hats ( A ) denote estimates, I t is the value for 

 abundance index t in fishing season y, Q t scales north- 

 ern anchovy biomass to the units of abundance in- 

 dex t, and p t is the relative contribution of a north- 

 ern anchovy at age a to abundance index t. We as- 

 sumed age-specific selectivity patterns for abundance 

 indices because estimates for most parameters were 

 available outside of the model. This approach gave a 

 more realistic model without increasing the number 

 of parameters estimated. Values of p t were relative 

 measures scaled to the interval [0,1], and the age 

 with maximum relative contribution for abundance 

 index t had p =1.0. 



Estimates of the scaling parameter for DEP data 

 (Q DEp =l) and age-specific parameters (p ta ) for DEP, 

 HEP, and EPI data were from Methot (1989). Two- 

 year-old northern anchovy are all sexually mature 

 during the peak spawning period (p D£P2+ =1.0), 

 whereas the fraction of one-year-olds that are ma- 

 ture (p DEP i ) depends on water temperatures. Matu- 

 rity of age-1 northern anchovy during the peak 

 spawning season was calculated from mean Janu- 

 ary-February sea surface temperatures at Scripps 

 Pier, San Diego, California (Table 1), as described in 

 Methot (1989). 



Estimates of age-specific egg production for ac- 

 tively spawning northern anchovy (Methot, 1989) 

 were used to estimate the age-specific parameters 

 (Phep.o and Pepia ) for egg production indices. No age- 

 northern anchovy spawn during the peak spawn- 

 ing period but all are actively spawning by age 2. 

 The fraction of actively spawning fish was also cal- 

 culated from mean sea surface temperatures (Methot, 

 1989). 



Age-specific parameters for contribution to egg 

 production indices (Phep.o and Pepia) were assumed 

 to be the product of relative egg production and frac- 

 tion active. Relative egg production values were the 

 same as those used by Methot (1989) and originally 

 by Parrish et al. (1986). 



For simplicity, relative age-specific contributions 

 to indices of schooling biomass (SPOTTER and SO- 

 NAR) for northern anchovy ages 1 and older 

 (Pspotter,i+ and Psonar,i+) were assumed to be 1.0. 

 The contribution of age-0 northern anchovy to the 

 SPOTTER and SONAR indices was estimated as 



-X^Xa^-^IX, 



(id 



Psi'DTTER.O ~ PSONAR.O 



10) 



1+e" 

 where jt was a parameter estimated by the model. 



Objective function 



Parameters in SMPAR were estimated by maximizing 

 a function proportional to the total log-likelihood (L lotal ): 



where N t is the number of observations for abundance 

 index t, and N is the number of recruitment esti- 

 mates. The k t values are weights that determine how 

 important different types of data are in parameter 

 estimation; they were set to one except during sensi- 

 tivity analyses. D t is the log-scale standardized re- 

 sidual for abundance index t in fishing season y and 

 R is the log-scale standardized residual for recruit- 

 ment in fishing season y: 



I) 



In (/,,,//,,,) 



£ t.y 



ln(/, v )-ln(/, v ) 



(12) 



R, 



-t.) 



\n(B Qy /B ) 

 a 



a 



(13) 



where e t is the log-scale standard error for abun- 

 dance type t in fishing season v, and a is the stan- 

 dard deviation for log-scale recruitment deviations 

 (8 in Eqn. 6). Log-scale standard errors for abun- 

 dance data (c ) were calculated from arithmetic scale 

 coefficients of variation by inverting Equation 1. 



The first term on the right side of Equation 11 gives 

 the log likelihood of abundance indices given param- 

 eters in the model. The second term gives the log 

 likelihood of recruitment estimates. Mean recruit- 

 ment ( B„ in Eqn. 13) is a "nuisance" parameter that 

 was set equal at each iteration to the mean of cur- 

 rent recruitment estimates. The log-scale standard 

 deviation assumed for recruitments (a=0.71) was 

 calculated from stock synthesis model 2 recruitment 

 estimates and was higher than the average standard 

 deviation (0.48) for 41 other stocks of clupeoid fishes 

 (Beddington and Cooke, 1983; Myers et al., 1990). 



The likelihood term for recruitments in Equation 

 11 is a constraint that penalizes individual recruit- 

 ment estimates that are different from the mean. 

 Larger deviations and smaller a values result in 

 larger penalties. The constraint does not penalize 

 serial correlation so that "runs" of good or bad re- 

 cruitments can be estimated by the model. This was 

 important because northern anchovy recruitments 

 tend to be serially correlated (see below). 



Jacobson and Lo 2 showed that a northern anchovy 

 model without age-composition data or a recruitment 



