Somerton et al,: Whole-gear efficiency of a benttiic survey trawl for flatfish 



281 



used to minimize the effects of the spatial variation in 

 fish density on catch. In each geographical block, three 

 nearby, but nonoverlapping, trawl hauls were made with 

 each of three bridle lengths (chosen in random order). 

 Bridles lengths measured 36.6 m, 54.9 m (the stan- 

 dard used on AFSC bottom trawl surveys), and 73.1 m. 

 Tailchain length was 6.1 m. Trawling was conducted 

 during daylight hours for 30 min at 1.5 m/sec. On all 

 hauls, door spread, wing spread, and headrope height 

 were measured simultaneously and continuously with 

 an acoustic trawl mensuration system. Tow length was 

 measured as the straight-line distance between the GPS 

 positions of the first and last footrope contact with the 

 bottom; this distance was determined by using a bottom 

 contact sensor (Somerton and Weinberg, 2001) attached 

 to the center of the footrope. The catch from each haul 

 was first sorted to species, weighed in the aggregate, and 

 then all flatfish were measured for TL in centimeters. 



Estimates of W^,, for the three bridle lengths were 

 calculated from the estimates of the length of bridle 

 contact with the bottom, L^„, provided by the bridle con- 

 tact experiment described in Somerton (2003). Although 

 the lengths of the bridles in this experiment were the 

 same as those in the herding experiment, the length of 

 the tailchains was 10.4 m longer because the length of 

 the tailchain extensions vary with the size of a vessel. 

 Consequently, the distance between the wing tip and 

 the door differed between experiments. Because the 

 cable used for the tail chain extension is quite similar 

 in diameter to that used for the bridles (i.e., 19 mm 

 [tail chain] vs. 16 mm [bridle]), the resulting differ- 

 ence in length is essentially the same as a constant 

 addition to the three bridle lengths. We assumed that 

 the effect of such a change in total bridle length was 

 reflected only in L^„, and that the portion of the bridle 

 that was off bottom, L^,^y, did not differ between the 

 bridle measurement and herding experiments. Thus L^^ 

 for the herding experiment was estimated as the total 

 bridle length (bridle length + tail chain length) for the 

 herding experiment minus L^,^y from the bridle contact 

 experiment. A value of W^^ was then estimated for each 

 bridle length as 



W„ =2sin(a)L„ 



(7) 



where a = the average bridle angle at each bridle length 

 during the herding experiment. 



Sin(a) was computed for each haul as 



iW,-W„)/2L„ 



where L, = the total length of the bridle plus the tail- 

 chain (i.e., wingtip to door distance); and 



Wy and W^ - the haul mean values of door and wing 

 spread. 



Variance of W^,„ was estimated by using the delta method 

 (Seber, 1973) and assuming no covariance between sintcO 

 and L . This variance is expressed as 



VariW^,,, ) = 4(sin(a)-yar(L„„ ) + L^^ Var(sin(a))). (8) 



Var(sin(a)) for each bridle length was calculated as 

 the between-haul variability in sin(a), and Var(L^„) was 

 obtained from Somerton (2003). 



Estimating h from the experimental data 



The herding coefficient was estimated by fitting a modi- 

 fied version of Equation 1 to the experimental data on 

 W,^, Wg„, and catch (in numbers). The first modifica- 

 tion, which is considered more fully in Somerton and 

 Munro (2001), consists of introducing a new parameter, 

 k, defined as the product of D and /?„, that is allowed to 

 vary among blocks. The second modification is to allow 

 length dependency in k and h. The modified equation is 



N,j, = k,,(LW„ ),j + k^K{LW,„ )„ + £,^„ 



(9) 



where subscript i refers to block number, j to bridle 

 length within block, k to fish length class, and f,j^ is a 

 normally distributed error term. 



For each fish-length class, fitting Equation 9 to the 

 herding data required estimation of n+\ parameters, 

 where n is the number of blocks sampled (i.e., a unique 

 value of k for each block and a common value of h 

 for all blocks). Because the model is nonlinear in the 

 parameters (h and k are multiplied together), it was 

 fitted to data by using nonlinear regression (Venables 

 and Ripley, 1994). Fish length classes used in the 

 calculations were chosen such that the number of ob- 

 servations of length was approximately equal among 

 classes, and differed among species due to differences 

 in the number and size range of sampled fish. After 

 /; was estimated, k^^ was calculated for the standard 

 bridle length with Equation 2.Variance of /jj, was es- 

 timated by using a bootstrapping process designed 

 to include among-block variability in catch and trawl 

 measurements as well as the uncertainty in the es- 

 timated value of W,„. First, a bootstrap replicate of 

 catch and trawl measurement data was obtained by 

 randomly choosing, with replacement, blocks of data, 

 each containing a single haul at each bridle length, 

 from the n blocks sampled (blocks rather than hauls 

 were randomized to preserve the within-block correla- 

 tion in catch). Second, for each bridle length, a value of 

 W „ was computed by using a normal random number 

 generator with values of the mean and variance of W„„ 

 reported in Somerton (2003). Third, h was then esti- 

 mated by fitting Equation 9 by using nonlinear regres- 

 sion, and k^ was estimated from h by using Equation 

 2. Fourth, the process was repeated 100 times and the 

 variance of ^^ was estimated as the variance among 

 the replicates. Although the model that we used allows 

 for length-dependent herding, herding may not be a 

 length-dependent process in all species. To determine 

 if k^ varied with fish length, the estimated length- 

 specific values of k^ were regressed on the midpoints 

 of the length intervals. 



