OPTIMAL ESTIMATION OF RAIN-RATE PROFILES FROM SINGLE-FREQUENCY RADAR ECHOES

The significant ambiguities inherent in the determination of a particu lar vertical rain intensity profile from a given time profile of radar echo powers measured by a downward-looking (spaceborne or airborne) r adar at a single attenuating frequency are well documented. Indeed, on e already knows that by appropriately varying the parameters of the re flectivity-rain rate (Z-R) and/or attenuation-rain rate (k-R) relation ships one can produce several substantially different rain-rate profil es that would produce the same radar power profile. Imposing the addit ional constraint that the path-averaged rain rate be a given fixed num ber does reduce the ambiguities but falls far short of eliminating the m. While formulas to generate all mutually ambiguous rain-rate profile s from a given profile of received radar reflectivities have already b een derived, there remains to be produced a quantitative measure to as sess how likely each of these profiles is, what the appropriate ''aver age'' profile should be, and what the ''variance'' of these multiple s olutions is. To do this, one needs to spell out the stochastic constra ints that can allow us to make sense of the words ''average'' and ''va riance'' in a mathematically rigorous way. Such a quantitative approac h would be particularly well suited for such systems as the planned pr ecipitation radar of the Tropical Rainfall Measuring Mission (TRMM). I ndeed, one would then be able to use the radar reflectivities measured by the TRMM radar to estimate the rain-rate profile that would most l ikely have produced the measurements, as well as the uncertainty in th e estimated rain rates as a function of range. Such an optimal approac h is described in this paper.