194 



Fishery Bulletin 90(1). 1992 



E 



I- 

 z 



Z 



o 

 o 



UJ 



cr 



0.40 

 0.35 

 0.30 

 0.25 

 0.20 

 0.15 

 0.10 

 0.05 



2 4 6 8 10 12 14 

 % TIME AT SURFACE 



B. 



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 

 0.40 | I I M I in t, V n ^ A M ^ s ^ K — r— n0.40 



0.5 1.0 1.5 2.02.53.0 3.54.0 4.5 

 % TIME AT SURFACE 



example, with a sampling time at the surface represent- 

 ing 7% of the total duration of the tow, and with a rate 

 constant of 0.15/m, the calculated abundance will be 

 140% of the real value. With eggs highly concentrated 

 near the surface, that is with high values of k, even 

 briefer towing times at the surface will still result in 

 severe bias. 



The degree of bias B was also computed for similar- 

 ily varying k and Ld, again with a=a = 0.305m, but 

 with maximum depth D increased to 200 m (Fig. 3B). 

 For a rate constant of 0.15/m as in the previous exam- 

 ple, but with towing time at the surface representing 



Figure 3 



Isopleths (in %) of the ratio of biased abundance to true abun- 

 dance per surface area of a theoretical population of fish eggs. 

 The isopleths were computed for variable rate constants k of 

 a negative exponential model describing the vertical distribu- 

 tion of the eggs, and for varying degree of oversampling sur- 

 face water during an oblique plankton tow, L^ , expressed as 

 percent of the total duration of the tow. In (A), the maximum 

 sampling depth was set at 50m; in (B), the maximum sam- 

 pling depth was set at 200 m. All other parameters equal in 

 (A) and (B). 



only 2% of the total duration (e.g., 12 seconds at the 

 surface for a total tow time of 10 minutes), the calcu- 

 lated abundance of eggs will again be 140% of the real 

 value. This somewhat counterintuitive result stems 

 from the fact that by sampling deep strata, the frac- 

 tion of the tow occurring in the stratum where eggs 

 are present is proportionally smaller. As a result, brief 

 times of oversampling at the surface have proportional- 

 ly more effect on the calculated abundances of eggs 

 than when the tow extends only to shallow strata. 



Effects of oversampling surface water 

 on variance of total abundance 

 and on mortality rate 



Effect on variance 



Since the length of dragging at the surface is likely to 

 vary (as a function of weather, crew handling of the 

 net, etc.), a variance will be introduced in the computa- 

 tion of the total abundance of eggs over the studied 

 body of water. It was assumed that the abundance of 

 eggs (5000 eggs/m-) was constant over the surface of 

 a theoretical body of water where numerical experi- 

 ments were carried out. Ten oblique tows were made 

 in this theoretical body of water, with L = 1000 m, 

 a = a = 0.564 m, and k = 0.2/m, which are convenient 

 values for illustrative purposes. A different length of 

 drag at the surface (Lp) was assigned randomly to 

 each tow; ten random numbers were multiplied by an 

 arbitrary length of 6 m and the resulting Lp's were 6 

 (x2), 12, 24 (x3), 30, 36, 48, and 54 m. Ten estimates 

 of abundance of eggs per unit surface area were 

 calculated, and the mean was 5994 eggslrrr, with 95% 

 confidence intervals of 5646-6342 eggs/m-. This illus- 

 trates that small and variable lengths of drag at the 

 surface bias the estimated abundance over the whole 

 body of water, and add a substantial margin of uncer- 

 tainty to the estimate of local abundance. 



In another numerical experiment, the abundance of 

 eggs was again assumed constant throughout at 5000 

 eggs/m^, except this time the degree of contagion 



