Lo et al : Modeling performance of an airborne lidar sur^/ey system for anchovy 



267 



of 0.676, and that the density offish schools/nmi^ in 

 a school group had a logarithmic mean of 3.91 and a 

 logarithmic variance of 0.51 based on data from Mac- 

 Call.'^ The number of schools within a school group is 

 the product of the area of the school group and the 

 density of schools within. Thus, the mean number of 

 schools was 10,274^ in a school group (Table 2). The 

 diameter of anchovy schools was generated from the 

 frequency distribution of the diameters of anchovy fish 

 schools in the Southern California Bight (Table 1). 



On average, there were 150,000 anchovy schools 

 in the Southern California Bight in the 1970s (Mais, 

 1974). In recent years, the population has decreased 

 to one fifth of that level ( Jacobson et al., 1994). In 

 the simulation, we constructed populations compris- 

 ing 80,000, 32,000, and 16,000 schools with an aver- 

 age biomass of 12 metric tons (t). At each population 

 level of anchovy, we simulated school gi'oups for nine 

 combinations of three school diameters and three 

 school densities, each with a multiplier of 0.5, 1, and 

 1.5 applied to both mean and standard deviation 

 of \n(school diameters) and of \n(density of schools) 

 respectively (Table 2). For example, for a population 

 of 32,000 schools, a multiplier of 0.5 applied to both 

 mean and standard deviation of In(diameter) (an 

 area of 9.45 nmi- or 32.34 km-) for a school gi'oup,^ 

 and a multiplier of 1.5 applied to the mean and stan- 

 dard deviation of ln( density) (625 schools /nmi^ or 

 182 schools/km^),'' would yield an average number of 

 5914 schools per school group and an average of six 

 school groups (32,000/5914) (Table 2). This popula- 

 tion was denoted as 32,000 (0.5,1.5). The encounter 

 probabilities for seven swath widths (1, 10, 50, 200, 

 500, 900, and 1600 m for a total of 63 (3 x 3 x 7) 

 sets of scenarios) were simulated (in computation, 

 numeral 1 was used to represent diameters less than 

 or equal to 1 m). For each of the three populations, 

 500 iterations were run for each of 63 sets. The mean 

 of the encounter probabilities from 500 runs was used 

 to estimate the mean encounter probability. 



For multiple swaths (n ), the probability (p,, „ ) that 

 at least one of the swaths intercepts schools is com- 

 puted as 



where p is computed from Equation 1. 



(2) 



■• The mean diameter is 14.25 nmi = expi2.319+0.676/2l and the 

 mean density offish schools is 64.39 schools/nmi- = expi3.91 + 

 0.51/2), the mean number of fish schools in a school group = 

 (14.25/21-64.39=10,274. 



^ [exp(2.319 X 0.5 + 0.676 x 0.5 x 0.5/2 )/2P x 3.1416 = 9.45 nmi-. 



** exp(3.91 X 1.5 + 0.51 x 1.5 x 1.5/2) = 625.62 schools/nmi^. 



Estimating the number of swaths needed in a survey 



Typically, the optimal sample size for a survey is 

 computed by minimizing the variance of the esti- 

 mate subject to a fixed cost. Because this informa- 

 tion was not available, we defined a desirable sample 

 size in terms of the minimum number of transect 

 lines or swaths needed to guarantee at least one pos- 

 itive sighting at an acceptable probability. There- 

 fore, from p,, in Equation 1, one can compute the 

 number of swaths in ) needed for a desired value of 



Py.n by using 



ln(l 



Pv.n 



ln(l- 



(3) 



Pv' 



Probability of detecting fish by depth with 

 signal-to-noise ratios (SNR) 



The signal level of a lidar system decays exponen- 

 tially with depth. The decaying signal of a single 

 pulse can be expressed by the equation 



S( 2 ) = So — —^ exp( -202 ), (4) 



Po 



