594 
Fishery Bulletin 95(3), 1997 
Appendix 1 : Calculation of "curved" 
correction factors 
This appendix describes, for the case where the path of 
the net during hauling is assumed to be curved, the calcu- 
lation of the correction factors that convert egg counts to 
egg density (eggs/m 2 ). 
For each plankton tow, we calculated the following: 
2 n (i 1 ), and P (; (f 2 ) ma y written as P nl , z nl , and P u2 , 
respectively. 
From the above assumptions, P_ vV P .„ z v and w x are 
known, as are C(z) and C'(z) for all 2 . Also, of course, P n2 = 
Calculating P n] 
1 the position of the net at the start of hauling; 
2 the position of the net at a series of equally spaced 
times during hauling; 
3 the flow of water through the net at each of these times; 
and 
4 a correction factor for each of these times. 
Finally, interpolation was used to calculate the correction 
factor when the net was at the mid-point of the depth layer 
associated with each egg stage. This was taken as the cor- 
rection factor for that egg stage at that plankton tow. 
First, the assumptions behind these calculations and 
some notation are defined. 
Assumptions and notation 
The calculations required the following assumptions: 
1 the vessel drifted at a constant velocity while shooting 
and hauling; 
2 the net dropped at a constant speed during shooting; 
3 the warp was always straight and decreased in length 
at a constant rate during hauling; 
4 the net mouth was always perpendicular to the warp; 
5 the water velocity varied only with depth (not with lon- 
gitude, latitude, or time); and 
6 the following are known exactly: 
a) the vessel position at the time of shooting and at 
the start and finish of hauling, 
b) the net depth and warp length at the start of haul- 
ing, and 
c) the water velocity profile. 
Two further assumptions are described in sections “Cal- 
culating P d ” and “Calculating the net path during hauling.” 
Let P n (f) and P () (£) be 3-dimensional vectors describing 
the position of the net and the vessel at time t, where time 
and position are measured in relation to the time and the 
vessel position when the net was shot, and where the three 
coordinates give distances, in meters, to the east, north, 
and downwards, respectively. (In this Appendix, vectors 
are underlined to distinguished them from scalars.) Let t 1 
and 1 2 stand for the times at the start and finish of hauling. 
Let the water velocity at depth 2 be described by the 
vector C(z), and the mean water velocity between the sur- 
face and depth 2 by C'(z). Denote the net depth and the 
warp length at time t by z n (t) and w(t), respectively (note 
that z n (t) is the depth coordinate of P n (t)). 
Where it is convenient, the symbols t 1 and t 2 will be re- 
placed by subscripts 1 and 2, so that, for example, P n {t x ), 
In calculating the position of the net at the start of haul- 
ing, it was assumed that, while the net was sinking prior 
to hauling, its horizontal velocity was the sum of the wa- 
ter velocity and some fraction c of the vessel velocity (the 
latter component being caused by drag from the warp). Thus, 
P nl — cP vl + t x Q\Z nl ) + Z nl , (Al) 
where 0 < c < 1, and Z nl is the 3-dimensional vector, 
(0,0, z nl ). Also, from assumptions 3 and 6, 
\Enl-E.vl\ = W l- (A2 > 
The value of c (and thus P nl ) can be calculated by solv- 
ing the simultaneous equations A 1 andA2. Geometrically, 
this is equivalent to finding a point of intersection of the 
horizontal line defined by Equation Al and the horizontal 
circle defined by Equation A2 (both are at depth z nl ). 
Substituting for P nl from Equation Al in Equation A2 
and expanding, we get the quadratic equation in (c — 1 ) 
(c-l) 2 |P„ 1 f + 2(c-l)t 1 P vl > 
(A3) 
C" ( Z n j ) + 1 2 |C_(2 n 1 )| + 2 2 j — uv\ — 0 , 
which is solved with the usual formula ( • denotes the vec- 
tor dot product). 
Where there were two solutions for c (at all but a few 
stations), the one using the negative square root was cho- 
sen because it was usually the one that satisifed the con- 
dition 0 < c < 1. 
For the few stations where Equation A3 had no real so- 
lution (i.e. the line and circle did not intersect), P nl was 
taken to be the point on the circle closest to the line. It 
may be shown that this P n l is given by 
P, 
+ Z nl 
(A4) 
where r is the radius of the circle and P vl is the point, on 
the surface vertically above the line, that is closest to the 
shot position, and is given by 
t x C’(z nl )»P vX 
P'i = hQ\z nx) - 
— „i 
(A5) 
Stations where Equation A3 did not have a real solu- 
tion, or where c did not satisfy 0 < c < 1, were assumed to 
be instances where the above assumptions did not hold. 
