Analytical Expressions for Doppler Spectra of Scatter from Hydrometeors Observed with a Vertically Directed Radar Beam

A generalized expression is derived for the correlation function of signals backscattered from hydrometeors observed with a vertically pointed radar beam in which particle size distribution, turbulence, and mean wind are not homogeneous. This study extends the previous work of Fang and Doviak by including the effects of particle size distribution on the measured Doppler spectrum. It shows the measured Doppler spectrum to be the volumetric mean of the weighted convolution of the normalized Doppler spectra associated with turbulence, mean wind, particle oscillation/wobble, and terminal velocity. Without particle oscillation/wobble, mean wind, and turbulence, the Doppler spectrum is the mirror image of the terminal velocity spectrum under the condition that the second-order effect of finite beamwidth can be ignored. This generalized Doppler spectrum reduces further to a previously derived expression if the particle size distribution, or equivalently reflectivity, is uniform. Provided there is a unique relationship between the particle’s terminal velocity and its effective diameter, the derived equations can be applied to scatterers consisting of ice particles as well as water droplets. This study derives the analytical expression for the Doppler spectrum of mean wind and also shows that if stationary homogeneous turbulence is the only contributor to spectral broadening, then the average of a large number of radar-measured Doppler spectra will be equal to the velocity probability density function of turbulence independent of the angular, range, reflectivity, and weighting functions.