Stochastic-analytic approach to the calculation of multiply scattered lidar returns

The problem of calculating the nth-order backscattered power of a laser firing short pulses at time zero into an homogeneous cloud with specified scattering and absorption parameters, is discussed. In the problem, backscattered power is measured at any time less than zero by a small receiver colocated with the laser and fitted with a forward looking conical baffle. Theoretical calculations are made on the premise that the laser pulse is composed of propagating photons which are scattered and absorbed by the cloud particles in a probabilistic manner. The effect of polarization was not taken into account in the calculations. An exact formula is derived for backscattered power, based on direct physical arguments together with a rigorous analysis of random variables. It is shown that, for values of n less than or equal to 2, the obtained formula is a well-behaved (3n-4) dimensionless integral. The computational feasibility of the integral formula is demonstrated for a model cloud of isotropically scattering particles. An analytical formula is obtained for a value of n = 2, and a Monte Carlo program was used to obtain numerical results for values of n = 3, . . ., 6.