FISHERY BULLETIN: VOL. 70. NO. 3 



PARAMETER ESTIMATION WITH 

 THE MURPHY METHOD 



Reference Values 



We used the generalized Murphy catch equa- 

 tion (Tomlinson, 1970) to analyze aged Catch 

 data. Gotshall (in press) provides estimates of 

 natural mortality and biomass based upon a 

 fishery independent randomized trawling scheme 

 ( Abramson, 1968) . Since the biomass estimates 

 are inherently negatively biased regardless of 

 the catchability of shrimp and the mortality es- 

 timates may deviate from the population pa- 

 rameters in either direction, we decided to choose 

 a natural mortality which would provide Murphy 

 Method biomass estimates of a magnitude simi- 

 lar to those obtained from the randomized 

 trawling scheme. 



An annual natural mortality coefficient of M 

 = 1.44 applied to all age groups yielded the ap- 

 propriate biomass estimates. This is within the 

 range of the M values given in Gotshall's (in 

 press) Table 6 and cannot be considered signifi- 

 cantly different from those estimates in view of 

 the sizes of the standard errors shown in his 

 Table 9. 



Constructing Catch Ratios 



Ratios of number caught in month t + 1 to 

 number caught in month i were calculated for 

 all age III catches, giving values useful for with- 

 in-season estimation of fishing mortality. To 

 estimate across the closed seasons, the ratio of 

 catch at age III in the first catch-month of 

 season t + 1 to catch at age 11 in the last catch- 

 month of season i was calculated. For example, 

 with 2 seasons and 3 months in each season, 

 the ratios computed by this scheme would be: 

 E(l)=C„,(2)/C„i(l);/2(2)=C„r(3)/C,„(2); 

 R(3)=C„r(4)/C„(3); R(4)-C„i(5)/C,„(4); 

 i2(5) =Ciii(6)/Ciii(5), where the catches used 

 represent monthly catches by age (Table 4) and 

 a closed season exists between months 3 and 4. 



An additional assumption is that the exploi- 

 tation rate (E) during the last month of each 

 season is equal for ages II and III. Thus in the 

 example, £"11(3) =£7111(3). This assumption is 



necessary to allow estimation across the closed 

 season. 



Using these ratios for age III within season 

 and age III to age II between seasons and assum- 

 ing various exploitation rates for the last month 

 of fishing in 1968, it was possible to make nu- 

 merous estimates of the exploitation rates at 

 age III during each month of fishing from 1955 

 to 1968. The Murphy method with backward 

 calculation (Tomlinson, 1970) was used. The 

 technique is similar to one used by Murphy 

 (1965, 1966), except that Murphy used years 

 instead of months, combined some age groups 

 within years, had no years without catches, and 

 did not treat year classes separately. 



The data were separated into catches from 

 year classes 1952 through 1967. Using the same 

 hypothetical example as before (Table 4), the 

 catch data can be put in the order Ci(l), Ci(2), 

 Ci(3), 0, C„(4), C„(5), C„(6), 0, C„i(7), 

 Cm (8), Cm (9). The catch ratios are computed 

 as Ci(2)/C,(l), C,(3)/Ci(2), 0, C„(4)/Ci(3), 

 C„(5)/C„(4), C„(6)/C„(5), 0, Cm(7)/C„(6), 

 Cm (8) /Cm (7), Cm (9) /Cm (8). Since these 

 catches all came from the same cohort, the 

 Murphy method can be used to estimate £"1(1), 

 £"1(2),..., £111 (8) , given that Em (9) is known. 

 The previous analysis of age III data gave esti- 

 mates of E at age III during the last month of 

 fishing in each season, and these were used as 

 starting values for backward calculation on each 

 year class from 1952 through 1965. It was nec- 

 essary in estimating E for the 1966 and 1967 

 year classes to choose values which gave an av- 

 erage population size in 1968 similar to the re- 

 sults obtained from fitting the Schaefer model. 



Additional Modifications and Assumptions 



Two additional assumptions fundamental to 

 the results are: (1) ages II and III were ex- 

 ploited at the same rate, on the average, over 

 the entire time period; (2) the catchability co- 

 efficient (q), computed as monthly catch-per- 

 eff'ort in weight divided by estimated average 

 population weight for the combined age groups 

 during the month, was reasonably constant over 

 the entire time period. In order to satisfy these 

 two assumptions, it was necessary to alter some 



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