Fast Lidar and Radar Multiple-Scattering Models. Part I: Small-Angle Scattering Using the Photon Variance-Covariance Method

A fast, approximate method is described for the calculation of the intensity of multiply scattered lidar returns from clouds. At each range gate it characterizes the outgoing photon distribution by its spatial variance, the variance of photon direction, and the covariance of photon direction and position. The result is that for an N-point profile the calculation is O(N) efficient yet it implicitly includes all orders of scattering, in contrast with the O(N(m)/m!) efficiency of models that explicitly consider each scattering order separately for truncation at m-order scattering. It is also shown how the shape of the scattering phase function near 180 may be taken into account for both liquid water droplets and ice particles. The model considers only multiple scattering due to small-angle forward-scattering events, which is suitable for most ground-based and airborne lidars because of their small footprint on the cloud. For spaceborne lidar, it must be used in combination with the wide-angle multiple scattering model described in Part II of this two-part paper.