Weinberg and Somerton: Variation in trawl geometry due to unequal warp length 



33 



eye pollock (Theragra chalcogramrna) lack a herding 



response to the doors, bridles and mudclouds of survey 



trawls? ICES J. Mar. Sci. 61:1186-1189. 

 2003. Bridle efficiency of a survey trawl for flatfish: 



measuring the length of the bridles in contact with the 



bottom. Fish. Res. 60:273-279. 

 Somerton, D. A., and P. T. Munro. 



2001. Bridle efficiency of a survey trawl for flatfish. Fish. 



Bull. 99:641-652. 

 Somerton, D. A., and R. S. Otto. 



1999. Net efficiency of a survey trawl for snow crab, 



Chionoecetes opilio, and Tanner crab, C. hairdi . Fish. 



Bull. 97:617-625. 

 Somerton, D. A., and K. L. Weinberg. 



2001. The affect of speed through the water on footrope 



contact of a survey trawl. Fish. Res. 53:17-24. 



Stauffer. G. 



2004. NOAA protocols for groundfish bottom trawl sur- 

 veys of the nation's fishery resources. NOAA. Tech. 

 Memo. NMFS-F/SPO-65, 205 p. Alaska Fisheries 

 Science Center, 7600 Sand Point Way N.E., Seattle, 

 WA, 98115. 

 Venables, W. N., and B. D. Ripley. 



1994. Modern applied statistics with S-plus. 462 p. 

 Springer-Verlag, New York, NY. 

 Weinberg, K. L., D. A. Somerton, and P. T. Munro. 



2002. The effect of trawl speed on the footrope capture 

 efficiency of a survey trawl. Fish. Res. 58:303-313. 

 Weinberg, K. L., R. S. Otto, and D. A. Somerton. 



2004. Capture probability of a survey trawl for red 

 king crab iParalithodes camtschaticus). Fish. Bull. 

 102:740-749. 



Appendix 1: Estimating headrope and footrope 

 shape when the warps differ in length 



If a trawl headrope has the same shape as a flexible 

 twine under a uniformly distributed load, then the shape 

 of the headrope can be approximated as a quadratic 

 (parabolic) function (Fridman, 1969; p. 84) as 



y = cx 



(1) 



where c is a constant controlling the shape (Fig. Al). 

 As the headrope is distorted by a differential in warp 

 length, not only does the value of c change, but the 

 headrope is displaced along the path of the parabola, so 

 that its center is no longer aligned with the vertex of the 

 parabola. A unique solution to the shape of the headrope 

 when it is distorted in this manner can be determined 

 from three types of data: the total headrope length (L), 

 the measured distance between the wing tips (W), and 

 the measured slope (tangent) of the parabola at the 

 center of the headrope (tan). The third quantity can be 

 obtained from the V (perpendicular to the footrope) and 

 U (tangential to the footrope) velocities measured by the 

 headrope speed sensor as the quotient U/V. With these 

 quantities, the solution can be obtained as follows. 



A small length interval measured along the headrope 

 can be expressed as 



X(tn) 



Figure Al 



Shape of a trawl headrope as described 

 by a parabola. The total length of the 

 headrope (shown with a solid line) is 

 equal to L. The measured width of the 

 trawl (shown with a dashed line) is 

 equal to W. The circle indicates the 

 center of the headrope where a speed 

 sensor is located. The speed sensor 

 measures water speed both perpendicu- 

 lar and parallel to the headrope. 



ds = (dy +dx 



(2) 



Which, after substitution of the derivative of Equation 

 1, is 



ds = [l + (2cx)^fdx. 



(3) 



The length of any segment of the headrope, measured 

 from the port end (->;/„„ ^r'' i^ then 



S= \[l + (2cx)^y ds. 



(4) 



A segment equal to the total length of the headrope 

 is obtained by integrating up to the starboard end 



The solution is approximated numerically in two stag- 

 es. First, for a trial value of c. Equation 4 is integrated 

 from trial values of Ji:,^,,^,^ up to the value of x at which 

 S=L/2 (i.e., -Vjjjj^^/p). The tangent at this position is then 

 evaluated as 2cx,. 



nldlr 



(based on the derivative of Eq. 

 1). This process is then repeated iteratively to find the 

 value of .T:,„„.j,r for the specified value of c at which the 

 calculated tangent equals the tangent value determined 

 from the headrope speed sensor. The value of -v, is 



