Optimal estimation of rain rate profiles from single-frequency radar echoes

It is well-documented that there are significant ambiguities inherent in the determination of a particular vertical rain intensity profile from a given time profile of radar echo powers measured by a downward-looking radar at a single attenuating frequency. Indeed, one already knows (Haddad et al., 1993) how to vary the parameters of the reflectivity-rainrate (Z-R) and attenuation-rainrate (k-R) relationships in order to produce several substantially different rain rate profiles which would produce the same radar power profile. Imposing the additional constraint that the path-averaged rain-rate be a given fixed number does reduce the ambiguities but falls far short of eliminating them. While the authors have derived the formulae to generate all deterministic mutually ambiguous rain rate profiles from a given profile of received radar reflectivities, there remains to produce a quantitative measure to assess how likely each of these deterministic profiles is, what the appropriate “average” profile should be, and what the “variance” of these multiple solutions is. Of course, in order to do this, one needs to spell out the stochastic constraints that can allows sense to be made of the words “average” and “variance” in a mathematically rigorous way. Such a quantitative approach would be particularly well-suited for such systems as the spaceborne Ku-band precipitation radar of the Tropical Rainfall Measuring Mission (TRMM). Indeed, one would then be able to use the radar reflectivities measured by the TRMM radar to estimate the rain rate profile that would most likely have produced the measurements, as well as the uncertainty in the estimated rain rates, as a function of range. This paper presents an optimal approach to solve this problem