268 



Fishery Bulletin 98(2) 



where z = depth in meters; 



Sq = the signal level at the surface; 



/3 , = the clear-water backscatter coefficient; 



/L = the backscatter coefficient of a school of 



fish; 

 ^Q = the backscatter coefficient at the sur- 

 face; and 

 a = the lidar attenuation coefficient. 



The backscatter coefficients, P, have units of 1/m and 

 represent the fraction of the energy that would be 

 scattered upward by a 1-m layer of either clear water 

 or fish. By clear water we mean natural sea water 

 with its attendant load of yellow substance, plank- 

 ton, silt, etc., but without fish. The lidar attenuation 

 coefficient is related to the absorption and scatter- 

 ing coefficients of the water, in a way that is not com- 

 pletely understood, but depends on the field of view 

 of the lidar. In an operational system, this parameter 

 can be obtained directly from the lidar data. A very 

 narrowly collimated system (defined as one where 

 the field of view is much smaller than the average 

 scattering angle in the water and much smaller than 

 the ratio of the beam attenuation coefficient to the 

 lidar height) will have an attenuation that is very 

 close to the sum of the absorption and scattering. 

 A wide field of view collects multiple scattered pho- 

 tons, and the attenuation is closer to the absorption 

 coefficient. 



The noise in a lidar system can come from sev- 

 eral different processes. One of these is likely to 

 predominate in any particular set of circumstances. 

 One source is thermal noise in the receiver. This is 

 an additive noise that is independent of the signal 

 level. It is Gaussian with a zero mean. Another 

 source of noise is the shot noise from the sum of the 

 signal current, background-light-generated current, 

 and detector dark current. This is a Poisson process 

 that depends on the total detector current. However, 

 except for very low illumination levels, the Poisson 

 distribution is nearly Gaussian, and we made this 

 approximation. Also, we noted that if the signal from 

 the fish school is very large, the detection probabil- 

 ity is nearly unity, and accurate modeling of the 

 noise distribution is not critical. If the fish signal 

 is small, the shot-noise variance will be very nearly 

 the same whether fish are present or not. This is the 

 situation that must be treated accurately, and so we 

 assumed that shot noise could be approximated by 

 an additive signal-independent Gaussian process for 

 the purposes of our study. The final noise source is 

 caused by variations of the optical properties of the 

 water with depth. Variations that are slow in com- 

 parison with the depth resolution of the lidar can be 

 estimated and eliminated. However, more rapid fluc- 



tuations would be indistinguishable from noise. In 

 the absence of a better model for these fluctuations, 

 we also assumed that they were Gaussian. Thus, 

 an additive signal-independent Gaussian noise was 

 considered, and the source of this noise was not 

 considered further. The final results would not be 

 very different if the dominant noise was not Gauss- 

 ian. Non-Gaussian noise would change the numeri- 

 cal values of the detection and false-alarm integrals. 

 Because of the strong exponential decrease in signal 

 level with depth, small changes in these values would 

 correspond to small changes in detection depth. A 

 similar effect was caused by our choice of threshold 

 level, which also changed the detection and false- 

 alarm integrals. We show that the results are not 

 very sensitive to our choice of threshold level for 

 the same reason. It is possible that the variations 

 in optical properties produce a highly non-Gaussian 

 noise that will have a significant effect, but we have 

 no evidence for this. 



The probability density function ( pdf) of the instan- 

 taneous signal (s) for a single pulse at some depth 

 can therefore be approximated by a normal pdf with 

 mean - S and variance =: d^. For illustration, we 

 assumed that a was not depth dependent, although 

 s clearly was. 



Detection was accomplished by setting a threshold 

 signal level above which we asserted that fish were 

 present. The detection probability is the probability 

 that the instantaneous signal is above this threshold 

 when fish are present (i.e. when /3. > 0). Thus, 



p{detection) = P(s>T)= 1-0 



r-s, 



o 



where T = the threshold level; 



S = a normal random variable with mean= 



S, and variance = a^; 

 Sr - the signal level with fish present; and 

 (p{u) = P(U<u) is normal distribution function 

 of U with mean = and variance = 1. 



Specifying that fish are present whenever the re- 

 ceived signal exceeds some threshold value entails 

 some probability of a "false alarm." This probability 

 can be calculated from 



Ptfalae alarm) = Pis >T)= 1-0 



T-S.,. 



o 



where S,^ = the signal from clear water. 



To reduce the number of free parameters, we nor- 

 malized everything by the noise level. Thus, we 

 defined a signal-to-noise ratio, SNR = (S^ - S^J/a 



