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A comprehensive review and extension of the theoretical bases for the measurement of the characteristics of rain and snow with vertically pointing Doppler radar are presented. The drop size distribution in rain can be computed from the Doppler spectrum, provided that the updraft can be estimated, but difficulties are involved in the case of snow. Doppler spectra and their moments are computed for rain by using various power law relations of fall speed υ versus particle diameter D and an exponential fit to the actual fall speed data. In the former case, there is no sharp upper bound to the spectra and all the spectral moments increase with rainfall rate R without limit; in the latter case, there is a sharp upper bound of the spectra corresponding to the limiting terminal velocity of raindrops, and the spectral moments approach an asymptote. Accordingly, the power laws are useful approximations over only limited ranges of precipitation rate. A comparison of theoretical and experimental mean Doppler velocity 〈υ〉 as a function of radar reflectivity factor Z shows that the empirical relation 〈υ〉 = 2.6Z0.107 of J. Joss and A. Waldvogel seems to be the only practical relation; even so, the scatter in 〈υ〉 is about ±1 m sec−1. This is also the kind of error to be expected in measuring updraft speeds by present methods. Such updraft errors result in unacceptably large errors in the drop number concentration estimated from Doppler spectra. In the absence of updrafts the mean Doppler velocity 〈υ〉 is uniquely related to Λ, the slope of the exponential drop size distribution. Simultaneous measurements of Z and 〈υ〉 can then be used to estimate N0, Λ, D0, M, and R, where N0 is the intercept of the exponential drop size distribution at D = 0, D0 is the median volume diameter, and M is the liquid-water content.