Lo et al : Modeling performance of an airborne lidar survey system for anchovy 



273 



data, suggesting that light was penetrating the first 

 layer. We also observed that the water returned from 

 below and above schools of fish and found that the 

 additional attenuation caused by the fish was small in 

 comparison with the background water attenuation. 



Proportion of fish schools detected (<7) 



The proportion offish schools detected in the upper 2 

 meters (q^) depends on the depth-specific probability 

 of detection (P„izy, Eq. 16) and the vertical distribu- 

 tion offish schools (Eq. 13). 



The quantity (gj was computed by numerical 

 integration: 



A. 



q, = \p„(ii)f(u)du 

 



=1 



(17) 



1-* 



2au + \n{TNR/A)-ij' 



(J 



A 



where p^(u) is derived from Equation 16 and flu) is 

 the exponential pdf derived from Equation 13: 



A(.) = -e- 



:i8) 



The quantity, q^, increases with depth z and reaches 

 an asymptote at2^^^{q;q<=l) andq is defined as the 

 proportion offish schools detected. 



Criterion for evaluating trade-offs between 

 penetration depth and swath width 



If laser power is held constant, an increase in swath 

 width would decrease the maximum depth of penetra- 

 tion of the laser pulse. In this section, we established 

 a criterion for comparing various instruments having 

 different combinations of swath width and laser power 

 (maximum penetration depth). The effectiveness of 

 the width of a swath (y ) can be measured by the prob- 

 ability that some fish schools will be encountered (p^,) 

 in the swath (Eq. 1 from simulation). The effective- 

 ness of a lidar in detecting schools within the swath 

 is measured by the proportion offish schools detected 

 iq) (Eq. 17). The product of p^, q (Eqs. 1 and 17) is 

 then used to evaluate the overall effectiveness of any 

 instruments with a given swath width (y). 



Results 



Effects of swath width on encounter probability 



We assumed that schools were aggregated into school 

 groups in the survey area (42,204 km') and that 



school diameters and densities were equal to, or less 

 than, those reported by Smith (1981). Our simula- 

 tion results indicated that swath width had little 

 effect on the probability of encountering schools. 

 This was true for all three population sizes: 16,000, 

 32,000, and 80,000 schools ( Fig. 4). Encounter proba- 

 ability was affected by the swath width only when 

 the diameters of the school groups were small and 

 the school density within the school group was so 

 low (both multipliers were 0.5) that their distribu- 

 tion became nearly random rather than aggregated. 

 In these cases, the encounter probability increased 

 sharply when the swath width increased from 1 

 m to 50 m. Even this very limited effect of swath 

 width diminished as the number of schools in the 

 survey area increased. The encounter probability for 

 swath widths greater than 50 m was almost con- 

 stant regardless of conditions. 



For the multiple swaths, the probability that at 

 least one of them would intercept anchovy schools 

 (Eq. 2) was high in general. The lowest probability 

 was 0.65, for the case where fish were aggregated in 

 few large school groups of low population, i.e. 16,000 

 (1.5,1.5) for /2=5 (Eq. 2). For n^lO, the probability 

 (p^, „ ) was close to one for all cases. 



Depth-specific detection probability 



The depth at which a lidar is capable of detecting 

 a school or target will depend in part on the thres- 

 hold setting of the instrument in relation to the noise 

 (TNR). To illustrate these relationships we fixed a 

 false-alarm rate iPifalse alarm ) for the detection of 

 schools, used an alarm rate to determine the thresh- 

 old level, and then calculated the detection proba- 

 bility for schools (P(detection)). The results of such 

 a calculation are presented in Figure 5, where the 

 detection probability for fish schools was plotted as a 

 function of the probability of a false alarm for signal- 

 to-noise ratios of 1 and 3. Zero, the lower limit of the 

 plot, corresponds to a very high threshold (TNR) set- 

 ting, where the probability of a false alarm and the 

 probability of detecting a school are both zero. We 

 concluded that fish are never present at a setting of 

 zero. The upper limit of Figure 5 corresponds to a 

 very low threshold setting, where Pifalse alarm ) and 

 Pidetection ) are both unity; at a setting of 1, we con- 

 cluded that fish are always present. 



If one selects a reasonable false-alarm rate and a 

 signal-to-noise ratio at the surface, one can calculate 

 the detection probability as a function of depth. This 

 was done for a false-alarm probability of l^c and a 

 lidar attenuation coefficient of 0.1/m, and the results 

 are plotted in Figure 2 for several values of the sur- 

 face signal-to-noise ratio. There are several interest- 



