NOTE D'Amours and Gr^goire: Analytical correction for oversampled Scomber scombrus eggs 



195 



near the surface was made variable, i.e., the rate con- 

 stant k varied randomly within bounds. Ten transects 

 were carried out through these distributions of eggs, 

 with L = 1000m, a = a = 0.564m, and the length of drag 

 at the surface (Lp) was held constant at 50 m. For 

 each tow, a random value was assigned to the rate con- 

 stant k: ten random numbers were multiplied by 0.1/m, 

 and added to 0.15. The resulting values were 0.15, 0.16, 

 0.17 (x2), 0.20 (x2), 0.21 (x2), 0.22, 0.23 (x2), and 

 the corresponding values of Nq were adjusted so that 

 No/k = 5000 eggs/m^. Ten estimates of abundance of 

 eggs per unit surface area were calculated, and the 

 mean was 6876/m-, with 95% confidence interval of 

 6713-7039 eggs/m^. This indicates that even when 

 maintaining a constant length of drag at the surface, 

 similar problems of bias and variance still arise when 

 the degree of contagion of the eggs varies. 



Effect on mortality rates 



Again for numerical experiments, the abundance of 

 eggs at a theoretical station was assumed to be 5000 

 eggs/m^, with No = 475/m3, and k = 0.095/m (i.e., 

 475/0.095 = 5000). An oblique tow with L= 1000 m, 

 D = 50m, LD = 50m, and a = a' = 0.564m, was made 

 through this concentration of eggs at time to, and the 

 biased abundance was calculated to be 5873/m2. For 

 the purpose of the demonstration, it was assumed that 

 the eggs suffered no mortality. Some time later at time 

 tj , the abundance of eggs was still the same, but they 

 were closer to the surface, with No = 950/m^ and k = 

 0.19/m (i.e., 950/0.19 = 5000). The same oblique tow in 

 this slightly rearranged concentration of eggs yielded 

 a biased estimate of abundance of 6806/m2, a relative 

 increase of nearly 16% compared with the value at 

 to, and an absolute bias of over 36%. In real situa- 

 tions, then, an increase in the degree of vertical con- 

 tagion of the eggs over a sampling period could lead 

 to an underestimation of the mortality rate if the sur- 

 face water is oversampled, or to an overestimation if 

 the degree of contagion decreases. 



Conclusion 



The oblique tow is a convenient method to obtain an 

 estimate of abundance of eggs over a body of water. 

 However, in actual operating conditions, it is rarely 

 possible to carry out an oblique tow without dragging 

 the net at the surface for some period of time, which 

 may introduce a large bias in the estimate of abun- 

 dance. The first practical recommendation to avoid 

 such bias is to evaluate the assumption that a brief drag 

 time at the surface will cause only a small bias in the 

 estimation of abundance. If eggs are equally distrib- 



uted over considerable depth, or concentrated in deeper 

 water, this assumption is valid. If not, action should 

 be taken to avoid dragging the net at the surface. If 

 this is impossible, a measure of the amount of over- 

 sampling at the surface, and of the rate constant k, 

 should be used to remove the bias from the data follow- 

 ing Eq. 11. 



In this study, the percent time of the tow spent at 

 the surface was used as a measure of the amount of 

 oversampling at the surface; this information is read- 

 ily recorded in the field. However, the constant k 

 describing the distribution of eggs had to be approx- 

 imated from data available in the literature. Ware and 

 Lambert (1985) reported values of k ranging from 0.1 

 to 1.1; they further indicated that variations in k were 

 related to the steepness of the thermal gradient in the 

 water column, the development stage of the eggs, and 

 the degree of wind-induced mixing. Data on mackerel 

 egg distribution by deLafontaine and Gascon (1989) in- 

 dicated a mean value k = 0.1, with the lowest values for 

 the most recently spawned eggs. Data on mackerel egg 

 distribution by Sette (1943) indicated a higher mean 

 value k = 0. 17, but with the highest values for the most 

 recently spawned eggs. Differences in mean values of 

 k as well as development-stage specific values may 

 result from differences in local wind conditions as well 

 as in differences in local water density. As discussed 

 by Sundby (1983), the shape of a vertical distribution 

 of mackerel eggs will be determined by the difference 

 of density between the egg and the surrounding water, 

 and by the degree of wind-induced mixing. The rela- 

 tionship reported by Sundby (1983) between wind 

 velocity and vertical eddy diffusivity coefficient of 

 mackerel eggs indicates that the rate constant k should 

 diminish as the state of the sea increases. The definite 

 application of the analytical correction proposed herein 

 will require more site-specific studies on the factors af- 

 fecting the vertical distribution of mackerel eggs and 

 determining the value of k. Nonetheless, the value 

 k = 0.15 used in this study is representative of realistic 

 conditions in the field, and can be considered as a con- 

 servative estimate of the degree of vertical contagion 

 of mackerel eggs. With more reliable values of k, the 

 simple correction procedure suggested in this study 

 could help increase the accuracy of biological 

 parameters based on data from fish egg surveys where 

 the technique of the oblique plankton tow has been 

 used. 



Acknowledgments 



Dr. D. Booth and Mr. P. Gagnon reviewed an early ver- 

 sion of the manuscript. This work followed a study by 

 D'Amours (1988) during which help was provided by 



