Kleiber and Pemn: Reappraisal of catch and stock status in U.S. North Pacific albacore fishery 



383 



.o 



E 



o 



Figure 2 



Comparison of original catch-per-effort time series (solid line) 

 with a new catch-per-effort time series (dashed line) for North 

 Pacific albacore. The original time series (CPE dd ) is the same 

 as the solid line in Figure 1. The new time series (CPE„ e „) 

 is calculated using Equation 2 and ignoring strata with no 

 effort. 



where a is a favoritism coefficient. When a equals 1, 

 the probability of a stratum of either type being visited 

 is the same, and there is no favoritism. But as a in- 

 creases, the chance for any high-abundance stratum is 

 increased at the expense of the chance for any low- 

 abundance stratum. Note that the sum of probabilities 

 over all strata, n^ + n h pj,, is equal to 1, so that it is 

 certain that a given unit of effort will land somewhere 

 in the fishing grounds. 



The probability that any particular stratum will re- 

 ceive a particular visit is riipi for low-density strata 

 and n h p h for high-density strata. The expected total 

 number of visits is e tot niPi to any of the low-density 

 strata and e tot n h p h to any of the high-density strata, 

 where e tot is the total effort expended by the fishery. 

 Therefore, 



TO 

 "O 



To 

 o 



JD 



(5 



Q. 



E 



C 



LU 

 Q_ 



o 



1980 



1970 



1980 



1982 



1984 1986 



YEAR 



1988 



1990 



Figure 3 



Detailed time series of stratified catch-per-effort for North 

 Pacific albacore calculated for every ten-day period using 

 Equation 2 and ignoring strata with no effort. 



E[Q] = e to t n, pi 4; E[C h ] = e t o t n h p h d h 



(5) 



where C\ and C h are the catches from all the low- 

 density and high-density strata, respectively, and d] 

 and d h are measured in CPE equivalents. 



The old CPE is the total catch over total effort. Its 

 expected value is thus, 



E[C,] + E[C h ] 

 E[CPE old ] = = n lPl d, + n hPh d h . (6) 



e tot 



To calculate the new CPE, we need the number of 

 strata in each area with at least one visit, vi and v h . 

 To get these, we will first derive the number of low- 

 and high-density strata that are missed altogether. The 

 probability that a particular stratum is missed by a par- 

 ticular effort unit is 1 — pi for low-density strata and 

 1 - p h for high-density strata. The probability that a 

 particular stratum is unvisited by any effort unit, that 

 is, missing in the data, is thus, 



