EDWARDS and KLEIBER: NONRANDOMNESS ON LINE TRANSECT ESTIMATES OF DOLPHINS 



an unrealistically large number of TVOD with 

 unrealistically complete coverage of the sim- 

 ulated area. Of the four hundred 1° squares in 

 our 1,200 X 1,200 nmi area, no more than six 

 went unsampled during any simulation. As a I'e- 

 sult, encounter rates in each square reflected 

 very accurately the true school density in each 

 square. Stratifying by these encounter rates, 

 deriving a different estimate of average school 

 density for each stratum, taking the weighted 

 average of these estimates, and then extrapolat- 

 ing this weighted average to the entire area pro- 

 duced quite accurate estimates of total school 

 abundance. 



The slight negative bias in the cases of a sim- 

 ple environment may be due to a curious effect 

 that was not obvious until we made a movie of 

 vessel and school movements generated by 

 TOPS for a simulation of the simple moving to- 

 pogi'aphy. It appears in this movie that vessels 

 tend to undersample the areas of highest density 

 in the center of the gi'adual peaks, because the 

 vessels encounter enough schools along the 

 periphery to keep them from turning into these 

 high density, central areas. Undersampling the 

 highest densities of course will lead to an under- 

 estimate of the average density per square and 

 thus to an underestimate of total abundance. 



This avoidance of peak centers did not occur 

 with the complex, static environment used in our 

 simulations, apparently because most of the 

 peak area in this topography occurred within 

 only a few squares (Fig. 2b). Vessel speed was 

 apparently sufficient to carry most vessels into 

 the highest density areas before the effects of 

 sightings caused the vessels to slow down. 



Stratification by Smoothed and 

 Interpolated Encounter Rate 



Given the apparent accuracy of estimates 

 derived under the stratification by raw en- 

 counter rate, it would seem irrelevant to proceed 

 to the more complicated and sometimes ineffec- 

 tive stratification by smoothed encounter rates. 

 However, real world tuna vessels never sample 

 the ETP as completely as the simulated vessels 

 sampled the TOPS environments. In most years, 

 fewer than half the vessels carry observers, the 

 fleet as a whole samples less than half the entire 

 ETP, and the sampling that is done tends to be 

 concentrated seasonally in variable geographic 

 areas (Buckland and Anganuzzi 1988). This 

 leaves many 1° squares unsampled. 



For management purposes, we cannot assume 



that squares with no effort contained no dol- 

 phins; therefore, we are left with the necessity of 

 estimating densities in those unsampled areas. 

 We have to fill in the holes somehow, so an inter- 

 polation method, either a more robust method 

 than used to date, or some new method, is re- 

 quired. 



In most of the TOPS simulations, Buckland 

 and Anganuzzi's (1988) smoothing and interpola- 

 tion routines worked quite well, with accuracy 

 rivaling that of the raw encounter rate stratifica- 

 tion. The very poor performance of the smooth- 

 ing algorithm in the case of a complex, static 

 environment, however, is troubhng because we 

 have no data from the field to determine whether 

 or not such topogi'aphies exist in the ETP. We 

 suspect that such topogi'aphies do exist because 

 the parameters used in the TOPS model were 

 chosen specifically to be reasonable. In partic- 

 ular, the distances between peaks were chosen 

 to bracket the apparent distances between clus- 

 ters of dolphin schools as indicated by sightings 

 from research vessels. ^^ Also, the movement 

 rates by vessels, schools, and topogi'aphy were 

 specifically selected to approximate observed 

 rates. 



The severe bias in the complex static case 

 arises owing to an interaction between the effec- 

 tive sampling frequency (in this case, 1° 

 squares), the peak topography, and the me- 

 chanics of the smoothing algorithm. The algor- 

 ithm works by calculating, for each square, a 

 smoothed encounter rate that is a weighted 

 average of encounter rates for all squares within 

 a radius of at least four squares. Thus the 

 smoothed rate in each square is affected by rates 

 across a diameter of at least eight 1° squares, or 

 a distance of at least 480 nmi (8 x 60 nmi). In the 

 case of the complex, static topogi'aphy, this mini- 

 mum distance is gi'eater than the distance be- 

 tween peaks (300 nmi). Also, the relatively pre- 

 cipitous peaks encompass only 3 or 4 squares and 

 are separated by low-density areas several 

 squares across. The smoothing algorithm tends 

 to "fill in" these low-density areas, elevating ap- 

 parent encounter rates in the intervening 

 squares and causing squares to be assigned to 

 strata out of proportion to the true densities of 

 dolphin schools in the squares. 



It is possible that the relatively precipitous 

 slopes of the peaks in the complex environment 



"R. S. Holt. Southwest Fisheries Center. National Mar- 

 ine Fisheries Service, NOAA, P.O. Bo.\ 271, La JoUa, CA 

 92038, pers. eommun. December 1987. 



871 



