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

Abstract

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 rate R than is available from radar reflectivity Z alone. To derive rainfall rates from these new parameters such as differential reflectivity ZDR and specific differential phase shift KDP and 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 variables No, Do, and μ providing 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 constants a and b in the empirical relationships Z = aRb with the values of a and b obtained when R and Z are derived by integrating the appropriately weighted gamma function. The relationships in common use both for inferring R from Z, ZDR, and KDP, and for developing attenuation correction routines have been derived from a best fit through the values obtained by cycling over these predicted ranges of No, Do, and μ. It is pointed out that this derivation of the predicted range of No, Do, and μ arises 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 and Z, ZDR, and KDP need to be reexamined. When more realistic drop shapes are also used, it may be that published relationships for deriving R from Z and ZDR are in error by over a factor of 2; a new equation is proposed that, in the absence of hail and attenuation, should yield values of R accurate to 25%, provided that ZDR can be estimated to 0.2 dB and Z is calibrated to 1 dB. Relationships of the form R = aKbDP, with b = 1.15, are in widespread use, but more realistic drop spectra and drop shapes yield a value of b closer to 1.4, similar to the exponent in Z–R relationships. In accord, although KDP has the advantage of insensitivity to hail, it may have the same sensitivity to variations in drop spectra as Z does. In addition, the higher value of the exponent b implies that the proposed use of the total phase shift to give the path-integrated total rainfall is also questionable. However, the consistency of Z, ZDR, and KDP in rain can be used to provide absolute calibration of Z to 0.5 dB (12%), and when it fails it indicates that hail is present, in which case a relationship of the form KDP = aR1.4 should be used. The technique should work at S, C, and X band, but, in all cases, paths should be chosen so that the total phase shift is not large enough to introduce significant attenuation of Z and ZDR.