Multiple-scattering lidar equation.

A multiple-scattering lidar equation is derived from a phenomenological representation of the scattering processes. The contributions are separated into the unscattered, singly scattered, and multiply scattered illumination of the scattering volume, a single backscattering reflection from that volume, and the unscattered and multiply scattered propagation back to the receiver. The equation is obtained in the form of analytic expressions that explicitly show the signal dependence on the extinction coefficient, the effective particle size, the range, and the receiver field of view. Consistent agreement is found with Monte Carlo calculations and published laboratory measurements. Numerical simulations demonstrate that measurements made at three or more fields of view can be inverted to solve for the extinction coefficient and the effective particle radius. The multiple scatterings taken into account in the proposed equation are the small-angle diffraction scatterings; the wide-angle scatterings caused by refraction and reflection are considered lost, except for one backscattering at an angle close to 180°. Consequently, the equation is applicable to cases in which the projection of the lidar receiver field of view on the cloud is of the order of the angular width of the diffraction peak of the phase function times the penetration depth into the cloud.