This paper describes a computationally efficient nearly optimal Bayesian algorithm to estimate rain (and drop size distribution) profiles, given a radar reflectivity profile at a single attenuating wavelength In addition to estimating the averages of all the mutually ambiguous combinations of rain parameters that can produce the data observed, the approach also calculates the n-ns uncertainty in its estimates (this uncertainty thus quantifies "the amount of ambiguity" in the "solution") The paper also describes a more general approach that can make estimates based on a radar reflectivity profile together with an approximate measurement of the path-integrated attenuation, or a radar reflectivity profile and a set of passive microwave brightness temperatures This more general "combined" algorithm is currently being adapted for the Tropical Rainfall Measuring Mission

Stochastic Filtering of Rain Profiles Using Radar, Surface-Referenced Radar, or Combined Radar-Radiometer Measurements

It is well known that there are significant deterministic ambiguities inherent in trying to determine the particular rain-rate profile that produced some given sequence of air- or spaceborne radar echo powers at a single attenuating frequency. For different combinations of radar data, formulas for the mutually ambiguous solutions are derived and the resulting ambiguities are quantified mathematically. When the given data consist of a single radiometer measurement together with a single-frequency set of range-gated echo powers, it is shown that several substantially different rain profiles can still realistically be considered solutions. On the other hand, if the data consist of a two-frequency set of echo powers, it is proven that the inversion problem generically has a unique solution.

Intrinsic Ambiguities in the Retrieval of Rain Rates from Radar Returns at Attenuating Wavelengths

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. >

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

There are significant inherent ambiguities when one tries to determine a particular vertical rain intensity profile from a given time profile of radar echo powers measured by a downward-looking (spaceborne or airborne) radar at a single attenuating frequency. In this paper, we quantify these ambiguities mathematically, and examine their effects on the performance of rain-rate retrieval algorithms initially proposed for use by the Precipitation Radar of the Tropical Rainfall Measuring Mission (TRMM).

Z. Haddad

Papers

Optimal Estimation of Rain Rate Profiles From Radar Returns at Attenuating Wavelengths