Backscatter and extinction in water clouds

Abstract

An approximate relation between the volume extinction coefficient σ e and backscatter coefficient σ b of atmospheric cloud at visible and near‐infrared wavelengths is derived. The relation is only weakly dependent on the size distribution of droplets and has the form σ e / σ b = (8π/ g ){1 + k −2/3 (〈 r 4/3 〉/〈 r 2 ) ‐ δ[ k 2 (〈 r 4 〉/〈 r 2 〉) + k 4/3 (〈 r 4/3; 〉〈 r 4 〉/〈 r 2 〉〈 r 2 〉)]} where the extinction efficiency is approximated by a complex‐angular‐momentum‐theory result and the parameters g and δ are determined by approximating a running mean of the backscatter gain by G ( x ) = g (1 + δ x 2 ) ( x is droplet size parameter and δ ≪ 1), k is the wave number, and 〈 r n is the nth moment of the droplet size distribution. To zero order the relation is linear and independent of the droplet size distribution σ e = [8π/ g (λ)]σ b , where g (λ) is a slowly varying function of wavelength. At a wavelength λ = 1.06 μm the relation is σ e (km −1 ) = 18.2 σ b (km −1 sr −1 ). Predictions made with this simple zero‐order approximation are in good agreement (within 50%) with Mie calculations of extinction and backscatter coefficients based on 156 measurements of cloud droplet spectra in cumulus and stratus type clouds. The linear σ e ‐ σ b ; relation is also in agreement with extinction and backscatter measurements made on laboratory‐generated fog droplet distributions. The relation suggests that visible or near‐infrared extinction coefficients in cloud of unknown type could be inferred from lidar backscatter measurements alone, without knowledge of the cloud droplet size spectra, barring complications arising from multiple scattering contributions to the lidar return.