660 



Fishery Bulletin 93(4), 1995 



and, by assuming that In R and C are normal, 

 p(ln R,C t ) may be obtained from standard texts: 



p(ln#,C ( ) 



2KO XnR o cl {l-r'*) 1  ( 20) 



sources cited in Bradford (1992), the sampling pe- 

 riod for the surveys providing mortality estimates 

 and then plotted In M versus In T in order to test 

 for the existence of a power law relationship between 

 these two variables (Fig. 1). The regression (Fig. 1) 

 yields In M = -0.991 In T + 0.776, or, equivalently 



exp 



-1 



2(1- r.7) 



' (AC,) 2 2r cl AC,AR (Alnfi) 2 ' 



a ci°\nR 



„2 



Equations 20 and 19 may be substituted into Equa- 

 tion 18 to obtain an expression for r' ci . 



The integrations in Equation 18 can be straight- 

 forwardly executed to show that 



CT ln/? 



r 2 i 1/2 



[exp(ffi„R)-lJ 



(21) 



It is evident from this expression that if o ]nB is small, 



then r' ci = r ci . 



An identical result holds for the coefficient of cor- 

 relation between R and In N n designated r,' u ; it is 

 given by 



Mnfl 



r 2 1 1/2 



[exp(cr lnfl )-lj 



(22) 



M=2.17T' 



(24) 



This apparent tendency of M and T~ l to covary may 

 stem from the existence of excluded regions of the 

 M , T~ l plane. If M is small, mortality will be de- 

 tectable only if sampling times are well separated, 

 implying that small M corresponds to large T. Simi- 

 larly, if M is large, the interval between samples 

 cannot be great, because the abundance will possi- 

 bly decline rapidly below the threshold of detectabil- 

 ity; thus, large M corresponds to small T. 



We can now use Equation 24 to obtain a relation 

 for the true variance of M, Var(M), by substituting 

 Equations 1 and 3 and then substituting Equation 

 24 into the result, with the outcome 



Var(M) = (0.15M 



0.2 



0A2af)M 2 . 



(25) 



Even for a given life history stage, there can be great 

 differences in the estimation error for abundance. 

 For the Peterman (1981) salmon smolt study, a £ = 

 0.08, whereas for the juvenile groundfish surveys 

 examined in Myers and Cadigan (1993, a and b), o £ 



The coefficient of correlation between R and N n des- 

 ignated r„" , can be found through a procedure analo- 

 gous to that employed in the calculation of r c ' . The 

 result is 



exp(r m (T lnR a ]nim) )-l 



[exp (crf n R ) - l] ' [exp (cr, 2 n( ni) ) - 1] 



1/2 



(23) 



In the limit that a ln(m) «l, Equation 23 reduces to 



Results 



Variability of mortality 



To obtain a relationship between the true variance 

 of mortality, Var(M), and mean mortality we can sub- 

 stitute Equation 3 into Equation 1. However, it must 

 be borne in mind that there is likely to be a relation- 

 ship between M and T (Taggart and Frank, 1990). 

 To address this problem, we extracted, from the 



