420 



Fishery Bulletin 90(2). 1992 



Determination of dolphin velocity 

 from range and bearing measurements 



In this section we relate the dolphin velocity to prac- 

 tical in situ measurements. We will derive a relation- 

 ship between dolphin speed and direction to its range 

 < r < oo and bearing < 6< 27i from the vessel that trig- 

 gers the dolphin to flight. 



For the times {tji i = l,2,...,n} we perform the cor- 

 responding measurements {ri,Gi. i = l,2,...,n}. It is not 

 necessary that the measurements be made from the 

 vessel, but with reference to it (e.g., aerial measure- 

 ments). However, it is necessary that there be no other 

 vessel in the vicinity that perturbs the measurements 

 by reaction of the dolphin to its presence. 



The measurements of range and bearing of the 

 dolphin from the vessel are equivalent to the cylindrical 

 coordinates of the dolphin with respect to the moving 

 frame of reference of the vessel. The cartesian coor- 

 dinates -oo<x<oo and -oo<y<oo with respect to the 

 same frame of reference are determined from 



X = r cos 9 and y = r sin 9. 



(19) 



Let Ax, Ay, and At be, respectively, the increments of 

 the variables x, y, and t. For each of the (n-1) con- 

 secutive intervals, we can compute the average speeds 

 in the x and y directions by 



Ax , ^, Ay 



V^ = — and Vv = — . 



At 



At 



(20) 



These are the components of the dolphins' apparent 

 velocity V in the frame of the moving vessel (Fig. 2). 

 We can express the dolphin velocity in the moving 

 frame as a function of its speed and direction in the 

 stationary frame by 



Vx = Vd cos a - Vb and Vy = Vq sin a, (21) 



which are a system of two coupled, nonlinear equations 

 with Vd and a as unknowns. The solution to this set 

 of equations is given by 



With these results we can compare the (n-1) time- 

 intervals the direction a taken by the dolphin given Eq. 

 (23), with the optimal direction given in Eq. (9) com- 

 puted from the result given Eq. (22). 



Acl<nowledgments 



We gratefully acknowledge the valuable suggestions 

 by Dave Au and Bill Perrin, and a correction of an 

 erroneous definition by Tim Gerrodette. 



Citations 



Au, D., and W. Ferryman 



1982 Movement and speed of dolphin schools responding to 

 an approaching ship. Fish. Bull., U.S. 80:371-379. 

 Au, D., and D. Weihs 



1980 At high speeds dolphins save energy by leaping. Nature 

 284(5756):548-560. 

 Burnham, K.P., D.R. Anderson, and J.L. Laake 



1980 Estimation of density from line transect sampling of 

 biological populations. Wildl. Monogr. 72, 202 p. 

 Hewitt, R.P. 



1985 Reaction of dolphins to a survey vessel: Effects on cen- 

 sus data. Fish. Bull., U.S. 83:187-193. 

 Perrin, W.F. 



1968 The porpoise and the tuna. Sea Frontiers 14:166-174. 



1969 Using porpoise to catch tuna. World Fishing 18(6): 

 42-45. 



Stuntz, W.E.. and W.F Perrin 



1979 Learned evasive behavior by dolphins involved in the 

 eastern tropical Pacific tuna purse seine fishery. In Abstracts 

 from presentations at the Third Biennial Conference of the 

 Biology of Marine Mammals, Seattle, Oct. 7-11, 1979, p. 

 58. [Avail. Library, Natl. Mar. Mammal Lab., Seattle 98115.] 



Vd = VV, + Vb + V/, (22) 



where we have chosen the positive root, and 



V„ 



a = arctan 



Vx + Vb 



(23) 



