The Need to Represent Raindrop Size Spectra as Normalized Gamma Distributions for the Interpretation of Polarization Radar Observations

Polarization radar techniques essentially rely on detecting the oblateness of raindrops to provide a measure of mean raindrop size and then using this information to give a better estimate of rainfall rateRthan is available from radar reflectivityZalone. To derive rainfall rates from these new parameters such as differential reflectivityZ DRand specific differential phase shiftK DPand to gauge their performance, it is necessary to know the range of naturally occurring raindrop size spectra. A three parameter gamma function is in widespread use, with the three variablesNₒ,Dₒ, andμproviding a measure of drop concentration, mean size, and spectral shape, respectively. It has become standard practice to derive the range of these three variables in rain by comparing the 69 published values of the constantsaandbin the empirical relationshipsZ=aRb with the values ofaandbobtained whenRandZare derived by integrating the appropriately weighted gamma function. The relationships in common use both for inferringRfromZ,Z DR, andK DP, and for developing attenuation correction routines have been derived from a best fit through the values obtained by cycling over these predicted ranges ofNₒ,Dₒ, andμ. It is pointed out that this derivation of the predicted range ofNₒ,Dₒ, andμarises using a flawed logic for a particular nonnormalized form of the gamma function, and it is shown that the predicted ranges give rise to some very unrealistic drop spectra, including many with high rainfall and very small drop sizes. It is suggested that attenuation correction routines relying on differential phase may be suspect and the commonly used relationships between rainfall rate andZ,Z DR, andK DPneed to be reexamined. When more realistic drop shapes are also used, it may be that published relationships for derivingRfromZandZ DRare in error by over a factor of 2; a new equation is proposed that, in the absence of hail and attenuation, should yield values ofRaccurate to 25%, provided thatZ DRcan be estimated to 0.2 dB andZis calibrated to 1 dB. Relationships of the form R = a K DP b , withb= 1.15, are in widespread use, but more realistic drop spectra and drop shapes yield a value ofbcloser to 1.4, similar to the exponent inZ–Rrelationships. In accord, althoughK DPhas the advantage of insensitivity to hail, it may have the same sensitivity to variations in drop spectra asZdoes. In addition, the higher value of the exponentbimplies that the proposed use of the total phase shift to give the pa