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

For a laser that fires a short pulse at time 0 into a homogeneous cloud with specified scattering and absorption parameters, this paper addresses the problem of theoretically calculating Jn(t), the nth-order backscattered power measured at any time t > 0. The backscattered power is assumed to be measured by a small receiver, which is colocated with the laser and which is fitted with a forward-looking conical baffle of adjustable opening angle. The approach taken here to calculate Jn(t) is somewhat unusual in that it is not based on the radiation-transfer equation but rather on the premise that the laser pulse consists of propagating photons, which are scattered and absorbed in a probabilistic manner by the cloud particles. Polarization effects have not been considered. By using straightforward physical arguments together with rigorous analytical techniques from the theory of random variables, an exact formula is derived for Jn(t). For n ≥ 2 this formula is a well-behaved (3n − 4)-dimensional integral. The computational feasibility of this integral formula is demonstrated by using it to evaluate Jn(t)/J1(t) for a model cloud of isotropically scattering particles; for that case an analytical formula is obtained for n = 2, and a Monte Carlo integration program is employed to obtain numerical results for n = 3, …, 6.