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Fishery Bulletin 98(2) 



"dropped" camera by day and Aoki and Inagaki 

 (1988) used a tethered camera by night and obtained 

 the packing density of anchovy with an average 

 length of 10 cm. 



The standard deviations ( SD ) of packing density of 

 anchovy computed from data collected by Aoki and 

 Inagaki ( 1988 ) and Graves ( 1977 ) measured only the 

 variation among schools and would underestimate 

 the overall variation of packing density. For this 

 reason, we used the coefficient of variation of packing 

 density of herring (1.55) (Misund, 1993; Loet al.-'), a 

 more realistic measurement of the variation of pack- 

 ing density, together with the mean packing density 

 of anchovy to estimate the mean (fi^) and variance 

 (a,-^) of log-transformed data:y = In(.v), where .v is the 

 packing density for 10-cm anchovy: 



a; =\n\cv'^{x)+ l], 



ft^. = \nix)-a;/2 



= ln(.v)-ln(cL'''^(x)-fl)/2, 



(14) 



(15j 



where cv{x) was that of herring (=1.55). Equations 14 

 and 15 were derived from the following two relations: 



;U,. = expifj^. +0^/2) and 

 a'^ =exp(2^,, -i-CTi^)[exp(cT^)-l]. 



Depth-specific probability of detection (pgCz)) 

 based on packing density 



As mentioned in an earlier section, P(c/e/ec</on) = 1 for 

 SNRz > TNR, and zero otherwise, because the steep 

 drop of Pidetection ) around z,„„^, the proportion offish 



that can be detected and identified at depth z, pjz), 

 was modeled by the PtSNR^ > TNR). The probability 

 of detection iPgiz)) was computed by means of the 

 probability density function of fish packing density 

 (.V) at depth z, assuming that SNR^^ is proportional to 

 the packing density ix), i.e. SNR^^ = Av, where A is 

 the proportionality and is a function of fish size and 

 reflectivity. If reflectivity is the same for all fishes, 

 then A is a function of fish size only, A ~ 10** x L-, 

 where L is the fish length in meters (Chumside et al., 

 1997 1. The packing density, x, is a lognormal random 

 variable. We could write SNR,-SNR^^ exp{-22a) = Av 

 exp(-23a); thus SNR^ > TNR is equivalent to x>(TNR/ 

 A)exp(2^a), and we approximated the proportion of 

 fish detected at depth z on the basis of the lognormal 

 distribution of packing density (x ) by 



p^{z}= p(x)dx 



e\pl2<t;:iTNR/A 



= p(ln(.Y) >[2az + \n(TNR /A)]) 

 I 2az + \n(TNR/ A)-iJ 



tl6j 



where <t>(u) = Pi U<u ) for the normal random vari- 

 able, [/, with mean = and variance = 

 1; and 

 In(.v) has meanw and variance d^. 



Equation 16 was computed through SNR,, the mean 

 of each individual normalized signal (or pulse). In the 

 appendix of this paper, we computed P„(z) through 

 individual normalized signals. We also assumed with 

 this computation that the effects of shadowing could 

 be neglected. Although more work on this issue is 

 needed, our results suggest that it is not a serious 

 effect. We observed multiple layers of fish in our 



