Fast Lidar and Radar Multiple-Scattering Models. Part II: Wide-Angle Scattering Using the Time-Dependent Two-Stream Approximation

Spaceborne lidar returns from liquid water clouds contain significant contributions from photons that have experienced many wide-angle multiple-scattering events, resulting in returns appearing to originate from far beyond the end of the cloud. A similar effect occurs for spaceborne millimeter-wave radar observations of deep convective clouds. An efficient method is described for calculating the time-dependent returns from such a medium by splitting the photons into those that have taken a near-direct path out to and back from a single backscattering event ( in the case of lidar, accounting for small-angle forward scatterings on the way, as described in Part I of this paper) and those that have experienced wide-angle multiple-scattering events. This paper describes the modeling of the latter using the time-dependent two-stream approximation, which reduces the problem to solving a pair of coupled partial differential equations for the energy of the photons traveling toward and away from the instrument. To determine what fraction of this energy is detected by the receiver, the lateral variance of photon position is modeled by the Ornstein-Furth formula, in which both the ballistic and diffusive limits of photon behavior are treated; this is considerably more accurate than simple diffusion theory. By assuming that the lateral distribution can be described by a Gaussian, the fraction of photons within the receiver field of view may be calculated. The method performs well in comparison to Monte Carlo calculations ( for both radar and lidar) but is much more efficient. This opens the way for multiple scattering to be accounted for in radar and lidar retrieval schemes.